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Lesson 1 Introduction
Outline Statistics Descriptive versus inferential statistics
Population versus Sample Statistic versus Parameter Simple Notation
Summation Notation Statistics What are statistics? What do you
thing of when you think of statistics? Can you think of some
examples where you have seen statistics used? You might think about
where in the real world you see statistics being used, or think
about how statistics in used in your major. Statistics are divided
into two main areas: descriptive and inferential statistics.
Descriptive statistics- These are numbers that are used to
consolidate a large amount of information. Any average, for
example, is a descriptive statistic. So, batting averages, average
daily rainfall, or average daily temperature are good examples of
descriptive statistics. Inferential statistics- inferential
statistics are used when we want to draw conclusions. For example
when we want to determine if some treatment is better than another,
or if there are differences in how two groups perform. A good book
definition is using samples to draw inferences about populations.
More on this once we define samples and populations. Population-
Any set of people or objects with something in common. Anything
could be a population. We could have a population of college
students. We might be interested in the population of the elderly.
Other examples include: single parent families, people with
depression, or burn victims. For anything we might be interested in
studying we could define a population. Very often we would like to
test something about a population. For example, we might want to
test whether a new drug might be effective for a specific group. It
is impossible most of the time to give everyone a new treatment to
determine if it worked or not. Instead we commonly give it to a
group of people from the population to see if it is effective. This
subset of the population is called a sample. When we measure
something in a population it is called a parameter. When we measure
something in a sample it is called a statistic. For example, if I
got the average age of parents in single-family homes, the measure
would be called a parameter. If I measured
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the age of a sample of these same individuals it would be called
a statistic. Thus, a population is to a parameter as a sample is to
a statistic. This distinction between samples and population is
important because this course is about inferential statistics. With
inferential statistics we want to draw inferences about populations
from samples. Thus, this course is mainly concerned with the rules
or logic of how a relatively small sample from a large population
could be tested, and the results of those tests can be inferred to
be true for everyone in the population. For example, if we want to
test whether Bayer asprin is better than Tylonol at relieving pain,
we could not give these drugs to everyone in the population. Its
not practical since the general population is so large. Instead we
might give it to a couple of hundred people and see which one works
better with them. With inferential statistics we can infer that
what was true for a few hundred people is also true for a very
large population of hundreds of thousands of people. When we write
symbols about populations and samples they differ too. With
populations we will use Greek letters to symbolize parameters. When
we symbolize a measure from a sample (a statistic) we will use the
letters you are familiar with (Roman letters). Thus, if I measure
the average age of a population Id indicate the value with the
Greek letter mu ( =24). While if I were to measure the same value
for a subset of the population or a sample then I would indicate
the value with a roman letter ( X =24). Simple Notation You might
thing about descriptive statistics as the vocabulary of the
"language" of statistics. If this is true then summation notation
can be thought of as the alphabet of that language. Notation and
summation notation is just a short hand way of representing
information we have collected and mathematical operation we want to
perform. For example, if I collect data on a variable, say the
amount of time (in minutes) several people spent waiting at a bus
stop, I can represent that group of numbers with the variable X.
The variable X represents all of the data that I collected.
Amount of Time
X 5.0 11.1 8.9 3.5 12.3 15.6
With subscripts I can also represent an individual data point
within the variable set we have labeled X. For example the third
data point, 8.9, is the X3 data point. The fifth data point X5 is
the number 12.3. Very often when we want to represent ALL of the
data
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points in a variable set we will use X by itself, but we may
also add the subscript i. Whenever you the subscript i, you can
assume that we are referring to all the numbers for the variable X.
Thus, Xi is all of the numbers in the data set or:
5,11.1,8.9,3.5,12.3,15.6. There are other common symbols we will
use besides X. Sometimes we will have two data sets to deal with
and refer to one distribution as X and the other distribution as Y.
It is also necessary for many formulas to know how many data points
are in a data set. The symbol for the number of data points in a
set is N. For the data set above the number of data points or N =
6. In addition, we will use the average or mean value a good deal.
We will indicate the mean, as noted above, differently for the
population () than for the sample ( X ). Summation Notation Another
common symbol we will use is the summation sign ( ). This symbol
does not represent anything about our data itself, but instead is
an operation we must perform. Whenever you see this symbol it means
to add up whatever appears to the right of the sign. Thus, X or Xi
tells us to add up all of the data points in our data set. For our
example above it would be: 5 + 11.1 + 8.9 + 3.5 + 12.3 + 15.6 =
56.4. You will see the summation sign with other mathematical
operations as well. For example X2 tells us to add all the squared
X values. Thus, for our example: X2 = 52 + 11.12 + 8.92 + 3.52 +
12.32 + 15.62 -or- 25 + 123.21 + 79.21 + 12.25 + 151.29 + 243.36 =
634.32. A few more examples of summation notation are in order
since the summation sign will be central to the formulas we write.
The following examples should give you a better idea about how the
summation sign is used. Be sure you recall the order of operations
needed to solve mathematical expressions. You will find a review on
the web page or you can click here:
http://faculty.uncfsu.edu/dwallace/sorder.html For the examples
below we will use a new distribution. X = 1 2 3 4 Y = 5 6 7 8
( )22 XX
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For this expression we are saying that the sum of the squared Xs
is not equal to the sum of the Xs squared. Notice here we want to
perform the operation in parentheses first, and then the exponents,
and then the addition. Thus:
( )22 XX ( )22222 43214321 ++++++
1 + 4 + 9 + 16 (10)2 30 100 For the next expression we show,
like in algebra, that the law of distribution applies to the
summation sign as well. Again, what is important is to get a feel
for how the summation sign works in equations.
YXYX +=+ )( (1+5)+(2+6)+(3+7)+(4+8) = (1+2+3+4)+(5+6+7+8) 6 + 8
+ 10 + 12 = 10 + 26 36 = 36
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Lesson 2 Scales of Measure
Outline Variables -measurement versus categorical -continuous
versus discreet -independent and dependent Scales of measure
-nominal, ordinal, interval, ratio Variables A variable is anything
we measure. This is a broad definition that includes most
everything we will be interested in for an experiment. It could be
the age or gender of participants, their reactions times, or
anything we might be interested in. Whenever we measure a variable,
it could be a measurement (quantitative) difference or a
categorical (qualitative) difference. You should know both terms
for each type. Measurement variables are things to which we can
assign a number. It is something we can measure. Examples include
age, height, weight, time measurement, or number of children in a
household. These examples are also called quantitative because they
measure some quantity. Categorical variables are measures of
differences in type rather than amount. Examples include anything
categorize such as race, gender, or color. These are also called
qualitative variables because there is some quality that
distinguishes these objects. Another dimension on which variables
might differ is that they may be either continuous or discreet. A
continuous variable is a variable that can take on any value on the
scale used to measure it. Thus, a measure of 1 or 2 is valid, as
well as 1.5 or 1.25. Any division on any unit on the scale produces
a valid possible measure. Examples include things like height or
weight. You could have an object that weighed 1 pound or 1.5 pounds
or 1.25 pounds. All are possible measures. Discreet variables, on
the other hand, can assume only a few possible values on the scale
used to measure it. Divisions of measures are usually not valid.
Thus, if I measure the number of television sets in your home it
could be 1 or 2 or 3. Divisions of these values are not valid. So,
you could not have 1.5 televisions or 1.25 televisions in your
home. You either have a television or you dont. Another way to keep
this difference in mind is that with a continuous variable is a
measure of how much. A discreet variable is a measure of how many.
Scales of Measure whenever we measure a variable it has to be on
some type of scale. The following scales are delivered in order of
increasing complexity. Each scale presented is in order of
increasing order. Nominal scales These are not really scales as
all, but are instead numbers used to differentiate objects. Real
world examples of these variables are common. The numbers
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are just labels. So, social security numbers, the channels on
your television, and sports team jerseys are all good examples of
nominal variables. Ordinal Scales Ordinal scales use numbers to put
objects in order. No other information other than more or less is
available from the scale. A good example is class rank, or any type
of ranking. Someone ranked at four had a higher GPA than someone
ranked as five, but we dont know how much better four is than five.
Interval Scales- Interval scales contain an ordinal scale (objects
are in order), but have the added feature that the distance between
scale units is always the same. Class rank would not qualify
because we dont know how much better one unit is than another, but
with interval there is the same distance from one unit to the next
anywhere we are on the scale. Examples include temperature (in
Fahrenheit or Celsius), or altitude. For temperature you know that
the difference in ten degrees is the same no matter how hot or cold
it might be. Ratio Scales Ratio scales contain an interval scale
(equal intervals between units on the scale), but have the added
feature that there is a true zero point on the scale. This zero
point is necessary for ratio statements to have meaning. Examples
include height or weight or measures of amount of time. Notice that
it is not valid to have a measure below zero on any of these
scales. Something could not weigh a negative amount. These scales
are much more common than interval scales because if a scale
usually has a zero point. In fact scientist invented the Kelvin
temperature scale so that they would have a measure of temperature
on a ratio scale. Again, in order to make ratio statements such as
something is twice or half of another then it must be a variable on
a ratio scale.
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Lesson 3 Data Displays
Outline Frequency Distributions Grouped Frequency Distributions
-class interval and frequency -cumulative frequency -relative
percent -cumulative relative percent -interpretations
Histograms/Bar Graphs Frequency Distributions We often form
frequency distributions as a way to abbreviate the values we are
dealing with in a distribution. With frequency distributions we
will simply record the frequency or how many values fall at a
particular point on the scale. For example, if I record the number
of trips out of town (X) a sample of FSU students makes, I might
end up with the following data: 0 2 5 3 2 4 3 1 0 2 6 0 4 7 0 1 2 4
3 5 4 3 1 6 1 0 5 3 Instead of having a jumbled set of numbers, we
can record how many of each value (f) there are for the entire
x-distribution. Below is a simple frequency distribution where the
X column represents the number of trips, and the corresponding
value for f indicates how many people in the sample gave us that
particular response. X f 0 5 1 4 2 4 3 5 4 4 5 3 6 2 7 1 From the
graph we can see that five people took no trips out of town, four
people took one trip out of town, four people took two trips out of
town, and so on. It is important not to confuse the f-value and the
x-value. The f-values are just a count of how many. So, you can
reverse the process as well. It might also be helpful in some
examples to go from a frequency distribution back to original data
set, especially if it causes confusion.
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In the following example I start with a frequency distribution
and go backward to find all the original values in the
distribution. X f 0 2 1 3 2 4 3 3 4 2 What is the most frequent
score? The answer is two because we will have four twos in our
distribution: 0 0 1 1 1 2 2 2 2 3 3 3 4 4 Grouped Frequency
Distributions The above examples used discreet measures, but when
we measure a variable it is often on a continuous scale. In turn,
there will be few values we measure that are at the exact same
point on the scale. In order to build the frequency distribution we
will group several values on the scale together and count any of
measurements we observe in that range for the frequency. For
example, if we measure the running time of rats in a maze we might
obtain the following data. Notice that if I tried to count how many
values fall at any single point on the scale my frequencies will
all be one. 3.25 3.95 4.61 5.92 6.87 7.12 7.58 8.25 8.69 9.56 9.67
10.24 10.95 10.99 11.34 11.59 12.34 13.45 14.53 14.86 We will begin
by forming the class interval. This will be the range of value on
the scale we include for each interval. There are many rules we
could use to determine the size of the interval, but for this
course I will always indicate how big the interval should be. In
the end, we want to construct a display that has between 5 and 15
intervals. Thus: Class Interval 0-2 3-5 6-8 9-11 12-14 Once we have
the class interval, we will count how many values fall within the
range of each interval. Since there is a gab in each class
interval, we will be actually counting any values that would get
rounded down or up into a particular interval. For example,
with
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the above data the value 8.26 would be rounded down into the 6-8
class interval. The value 8.69 would be rounded up into the 9-11
class interval. We will include a column to indicate the real
limits of the class interval. These are the limits of the interval,
including any rounded values. Real Limits Class Interval f -.5-2.5
0-2 0 2.5-5.5 3-5 3 5.5-8.5 6-8 5 8.5-11.5 9-11 7 11.5-14.5 12-14 5
Notice that my real limits cover half the distance of the gap
between each class interval. Most of the time this value will be
0.5 since most scales will have one unit values and 0.5 is half the
distance. So, real limits have no gap, but the class intervals do.
If a value falls exactly on one of the real limits we could
randomly choose its group. Cumulative Frequency Once we have formed
the basic grouped frequency distribution above, we can add more
columns for more detailed information. The first of these is the
cumulative frequency column. With this column we will keep a
running count of the frequency column as we move down the class
interval. Real Limits Class Interval f Cum. f -.5-2.5 0-2 0 0
2.5-5.5 3-5 3 3 5.5-8.5 6-8 5 8 8.5-11.5 9-11 7 15 11.5-14.5 12-14
5 20 So, at the first interval we have zero frequency, so
cumulatively we have zero values. For the second interval we have
three, so cumulatively we have three. For the third interval we
have five values, so cumulatively we have 8. That includes the five
for the third interval, plus the three from the previous intervals.
We continue this process until the last interval. Notice that when
we reach the last interval we have all the values in the
distribution represented. So, the bottom cumulative frequency is N
or the total number of values in the distribution (20 here).
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Relative Percent Another column will tell us the proportion of
total values that fall at each interval. That is, we will express
the frequency (column) as a percentage of the total. To convert the
frequency to a percentage take the frequency (f) and divide by the
number of values (N). This will give us the proportion of values
for that particular interval. Move the decimal over two places (or
multiply by 100) to change the proportion into a percent. Thus:
Real Limits Class Interval f Cum. f Rel % -.5-2.5 0-2 0 0 0 2.5-5.5
3-5 3 3 15 5.5-8.5 6-8 5 8 25 8.5-11.5 9-11 7 15 35 11.5-14.5 12-14
5 20 25 Cumulative Relative Percent For a final column we will keep
a running count of the relative percent column in the same way we
did with the cumulative frequency. Keep in mind we are counting
relative percentages now as we move down the display. Real Limits
Class Interval f Cum. f Rel % Cum. Rel. % -.5-2.5 0-2 0 0 0 0
2.5-5.5 3-5 3 3 15 15 5.5-8.5 6-8 5 8 25 40 8.5-11.5 9-11 7 15 35
75 11.5-14.5 12-14 5 20 25 100 Notice that we can keep a running
count of the relative percent column, but we could also obtain the
same numbers by computing the percentage for each cumulative
frequency as well. Interpretations The data display gives a good
deal of information about where values in the sample fall. One good
piece of information is about percentiles. A percentile is the
percentage at or below a certain score. You often get percentile
information when you get your SAT or ACT test scores back.
Percentile information is found in the cumulative relative
percentage column. Each value in that column tells us the
percentage of the distribution at that point or less on the scale.
Since we will be rounding values down into a certain interval based
on the real limits, then we will indicate where on the scale a
certain percentile is based on its corresponding upper real limit.
For example, what score corresponds with the 75th percentile? The
answer is 11.5 because any values of 11.5 or less are within the
bottom 75% of the distribution. Similarly, what percentile is
associated with a score of 8.5? We would use the cumulative
relative percent that corresponds to 8.5, which is 40%. So, the
score 8.5 corresponds with the bottom 40% or 40th percentile of the
distribution. Other interpretations from the table can be made as
well. For example, we might be interested in how many people fall
at a particular interval, or at or below a certain
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interval. How many scored between 3 and 5? The answer is a found
in the frequency column, or three. How many scored 8.5 or less? The
answer for this question is in the cumulative frequency column, or
eight. Histograms/Bar Graphs We can also take the frequency
information in our frequency or grouped frequency distribution and
form a graph. In the graph we will form a simple x-y axis. On the
x-axis we will place values from our scale, and on the y-axis we
will plot the frequency for each point on the scale. For grouped
frequency distributions, we will use the midpoint of each interval
to indicate different points on the scale. We will continue with
our previous example, but notice I have created a new column that
indicates the center or midpoint of each interval. We will use this
value to graph the display. Real Limits Class Interval MP f Cum. f
Rel % Cum. Rel. % -.5-2.5 0-2 1 0 0 0 0 2.5-5.5 3-5 4 3 3 15 15
5.5-8.5 6-8 7 5 8 25 40 8.5-11.5 9-11 10 7 15 35 75 11.5-14.5 12-14
13 5 20 25 100
Note that the bars are touching. The bars touch like this when
we are dealing with continuous data rather than discreet data. When
the scale measures discreet values we call it a bar graph, and the
lines do not touch. For example, if I measured the number of
democrats, republicans, and independents in a sample, we would use
a bar graph if we wanted to create a data display.
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0100
200
300
400
500
600
Dem Rep Ind
Party
Frequency
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Lesson 4 Measures of Central Tendency
Outline Measures of a distributions shape -modality and skewness
-the normal distribution Measures of central tendency -mean,
median, and mode Skewness and Central Tendency Measures of Shape
With frequency distribution you can an idea of a distributions
shape. If we trace the outline of the edges of the frequency bars
you can idea about the shape.
From this point on, I will draw these shapes to illustrate
different point throughout the semester. Keep in mind what you are
looking at is a line indicating the frequency or how many values in
a distribution lie at a particular point on the scale: just like a
histogram.
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Modality measures the number of major peaks in a distribution. A
single major peak is unimodal, and is the most common modality. Two
major peaks is a bi-modal distribution. You could also have
multi-modal distributions.
Skewness measures the symmetry of a distribution. A symmetric
distribution is most common, and is not skewed. If the distribution
is not symmetric, and one side does not reflect the other, then it
is skewed. Skewness is indicated by the tail or trailing
frequencies of the distribution. If the tail is to the right it is
a positive skew. If the tail is to the left then it is a negatively
skewed distribution. For example, a positively skewed distribution
would be: 1, 1, 2, 2, 2, 3, 3, 3, 9, 10. The outliers are on the
high end of the scale. On the other hand a negatively skewed
distribution might be: 1, 2, 9, 9, 9, 10, 10, 10, 11, 11, 11. Here
the outliers are on the low end of the scale.
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The normal distribution is one that is unimodal and symmetric.
Most things we can measure in nature exhibit a normal distribution
of values. Regression toward the mean is an idea that states values
will tend to cluster around the mean with few values toward the
trailing ends or tails of the distribution. As a result, most
things we measure will tend to have a normal shape. Think about
measures of height. There are very few people that are extremely
tall or extremely short, but most tend to cluster around the
average. With I.Q. scores, measures of weight, or most anything we
can measure, the same pattern will repeat. Since most things we
measure have more values close to the mean, we end up with mostly
normally shaped distributions.
Measures of Central Tendency Knowing where the center of a
distribution is tells us a lot about a distribution. Thats because
most of the scores in a distribution will tend to cluster about the
center. Measures of central tendency give us a number that
describes where the center lies (and most scores as well). Mean The
mean, or average score, is the arithmetic center of the
distribution. You can find the mean by adding all the scores ( X )
together and dividing by the number of values you added together
(N). X 1 2 3 4 5 15=X N = 5
For the Population: = XN
= 5
15 = 3
For the Sample: x = Xn
= 5
15 = 3
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Note that we calculate the mean the same way for both the sample
and the population we symbolize them differently. Other statistics
will differ in how they are computed for the sample versus the
population. Most students are familiar with these measures of
central tendency, but there are several properties that may be new
to you. 1) The first property of the mean is that it is the most
reliable and most often used measure of central tendency. 2) The
second property of the mean is that it need not be an actual score
in the distribution. 3) The third property is that it is strongly
influenced by outliers. 4) The fourth property is that the sum of
the deviations about the mean must always equal zero. The last two
properties need further explanation. An outlier is an extreme
score. It is a score that lies apart from most of the rest of the
distribution. If there are several outliers in a distribution it
will often result in skewed shape to the distribution. Outliers
tend to pull central tendency measures with them. Thus a
distribution of values 1, 2, 3, 4, 5 has an average of 3. Three
does a good job of describing where most of the scores in this
distribution lie. However, if there is an outlier, say by
substituting 25 for the 5 in the above distribution, then the mean
changes a great deal. The new distribution 1, 2, 3, 4, 25 has a
mean of 7. Seven is not really close to most of the other values in
the distribution. Thus, the mean is a poor measure of the center
when we have outliers, or a skewed distribution. A deviation is
just a difference. A deviation from the mean is the difference
between a score and the mean. So, when we say the sum of the
deviations about the mean must always equal zero is just a way of
saying that there are just as many differences between values above
the mean and the center as there are differences between values
below the mean and the center. Thus, for a simple distribution 1,
2, 3, 4, 5 the average is 3. Lets use population symbols and say =
3. The deviations are the differences between the score and the
mean. X X- 1 1-3 = -2 2 2-3 = -1 3 3-3 = 0 4 4-3 = 1 5 5-3 = 2 = 3
)( X = 0 Now if we add these deviations we will always get zero, no
matter what original values we use. This concept will be important
when we consider standard deviation because we will need to look at
differences between values in our distribution and the mean. Median
The median is the physical center of the distribution. It is the
value in the middle when the values of the distribution are
arranged sequentially. The distribution:
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1, 1, 2, 2, 3, 4, 5, 5, 5, 6, 7 has a median value of 4 because
there are five values above this point and five values below this
point in the distribution (1, 2, 2, 3, 4, 5, 5, 5, 6, 7). If you
have an even set of numbers then there will be two values that are
at the center, and you average these two values together in order
to determine the median. For example, if we take out one of the
numbers in the distribution so that we have 1, 1, 2, 2, 3, 4, 5, 5,
5, 6 then the two values in the center are 3 and 4 (1, 1, 2, 2, 3,
4, 5, 5, 5, 6). The average is 3.5 and that is the median. The
median is resistant to outliers. That is, outliers will generally
not affect the median and it will not be affected as much as the
mean. It is possible the median might move slightly in the
direction of the skew or outliers in the distribution. Mode The
mode is the most frequent value in the distribution. It is simply
the value that appears most often. For example in the distribution:
1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 5, 6, 7 there is only one mode (4).
But, in the distribution: 1, 1, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 6
there are two modes (3 and 4). If there is only one of each value
then there is no mode. The mode is not affected by outliers. Since
it is only determined by one point on the scale, other values will
have no effect. Skewness and Central Tendency We have already
discussed how each measure is affected by outliers or skewed
distribution. Lets consider this information further. In a
positively skewed distribution the outliers will be pulling the
mean down the scale a great deal. The median might be slightly
lower due to the outlier, but the mode will be unaffected. Thus,
with a negatively skewed distribution the mean is numerically lower
than the median or mode.
The opposite is true for positively skewed distributions. Here
the outliers are on the high end of the scale and will pull the
mean in that direction a great deal. The median might be slightly
affected as well, but not the mode. Thus, with a positively skewed
distribution the mean is numerically higher than the median or the
mode.
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Lesson 5 Measures of Dispersion
Outline Measures of Dispersion - Range - Interquartile Range -
Standard Deviation and Variance Measures of Dispersion Measure of
central tendency give us good information about the scores in our
distribution. However, we can have very different shapes to our
distribution, yet have the same central tendency. Measures of
dispersion or variability will give us information about the spread
of the scores in our distribution. Are the scores clustered close
together over a small portion of the scale, or are the scores
spread out over a large segment of the scale? Range The range is
the difference between the high and low score in a distribution.
Simply subtract the two numbers to find the range. So, in the
distribution: 1, 3, 5, 9, 11 the range is 11 1 = 10. Remember to
subtract the two numbers to give one number for the final answer.
Interquartile Range (IQR) The interquartile range is the range of
the middle 50% of a distribution. Because any outliers in our
distribution must be on the ends of the distribution, the range as
a measure of dispersion can be strongly influenced by outliers. One
solution to this problem is to eliminate the ends of the
distribution and measure the range of scores in the middle. Thus,
with the interquartile range we will eliminate the bottom 25% and
top 25% of the distribution, and then measure the distance between
the extremes of the middle 50% of the distribution that remains. To
actually compute some IQRs we would need to use calculus. Instead,
of that possibility we will use a method that will yield a
consistent, and somewhat accurate answer. Before we compute the
value, lets learn some new definitions. A quartile is a quarter or
25% of the distribution. When we compute the IQR we will want to
find each of the quartiles. The first quartile is the same as the
25th percentile because 25 percent of the distribution is at or
below that point. The second quartile is the same thing as the 50th
percentile and the median. The third quartile is the same as the
75th percentile. The IQR is the found by eliminating the values
that lie between the bottom end and the first quartile (bottom
25%). We will also eliminate the values between the third quartile
and the top of the distribution. We then subtract the new low and
high score of the left over middle part of the distributions. So,
IQR = Quartile 3 Quartile 1 or IQR = 75th percentile 25th
percentile. To compute the IQR first arrange your numbers from
lowest to highest and 1) find the median. The median is the 50th
percentile and second quartile. Its a starting point for us to find
the other quartiles. 2) Next find the median of the bottom half of
the distribution
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(ignoring the top half). This value is the 25th percentile or
first quartile because we have taken the bottom 50% and cut it in
half. 3) Find the median of the top half of the distribution just
like we did for the bottom. This value is the 75th percentile or
third quartile. 4) Next subtract the upper and lower medians you
found in step 2 and 3. In the following example the median is 8
because it is the average of the two middle numbers. The value 8 is
the 50th percentile or second quartile, though we will not use this
number in the computation.
1 2 5 6 7 9 10 12 15 19 Once we find the median we can divide
the distribution into two halves
1 2 5 6 7 9 10 12 15 19 Bottom 50% Top 50% The median of the
bottom 50% is 5. The median of the top 50% is 12. So, IQR = 12 5 =
7
Bottom 25% Top 25%
1 2 5 6 7 9 10 12 15 19
Standard Deviation and Variance While the interquartile range
eliminates the problem of outliers it creates another problem in
that you are eliminating half of your data. Generally, this is not
acceptable because of the difficulty in collecting data in the
first place. The solution to both problems is to measure
variability from the center of the distribution. Both standard
deviation and variance measure how far on average scores deviate or
differ from the mean. In this way, we use all the values in our
data to compute variability, and outliers will not have undue
influence.
Middle 50%
-
To compute standard deviation and variance we first start by
finding the deviation about the mean. Recall that we did the same
thing when discussing properties of the mean. Ill use the same
example with the simple distribution 1, 2, 3, 4, 5. First we find
the mean and the deviations about the mean. What we want to do is
add up these deviations and find out how far on average the scores
deviate from the mean. The problem we run into is that whenever we
add the deviations (in order to find the average of the deviations)
they will always sum to zero. How can we get an average if the sum
is always zero? X X- 1 1-3 = -2 2 2-3 = -1 3 3-3 = 0 4 4-3 = 1 5
5-3 = 2 = 3 )( X = 0 One solution is to square all of the
deviations. When we square all the numbers the negative values will
all become positive and we can then add the deviations without
getting zero. X X- (X )2 1 1-3 = -2 4 2 2-3 = -1 1 3 3-3 = 0 0 4
4-3 = 1 1 5 5-3 = 2 4 = 3 )( X = 0 2)( X = 10 Once we add the
squared deviations we have a measure of overall variability in the
distribution. The sum of the average squared deviations is called
the sums of squares, and will be used in almost everyone formula we
learn this semester. Please refer back to this section if formulas
give you problems later on in the course. Once we divide these
squared sums we will get the average squared deviation or variance.
In this example it is 10/5 = 2. Since we are in squared units and
not the same units as our scale we can take the square root of the
variance in order to get the standard deviation. The standard
deviation is the average deviation about the mean. For our example
we take the square root of 2 and find 1.41 is the standard
deviation.
-
The formula that contains all these operations is as follows.
Note that 2 is just the symbol we use for population variance and
is the symbol we use to denote population standard deviation.
( )22N
= X
= 2 population variance population standard deviation When
dealing with a sample a minor change to the formula is made, and
instead of subtracting the numerator by N, we divide by n 1. Try
the numbers in the above example to compute the sample variance and
standard deviation (variance is 2.5, standard deviation is
1.58).
( )221=
nXX
S
s = s2 sample variance sample standard deviation Please review
the animated demonstration on variance and standard deviation for
another example of how the population formula works. In addition an
alternative formula for these same computations is presented.
Although the formula detailed here is the best for understanding
the concept, the one presented on the web page will be easier to
use in the long run. Both appear in the homework packet formula
section as well. See
http://faculty.uncfsu.edu/dwallace/ssandrd1.html
-
Lesson 6 Z-Scores
Outline Linear Transformation -effect of addition/subtraction
-effect of multiplication/division Z-Transformation Z-Distribution
-properties Using Z-scores to compare values Linear Transformation
Anytime we change a distribution by using a constant we perform a
linear transformation. For example if I measure the heights of
everyone registered in this course, but then found the tape measure
I was using was missing the first two inches (so it started at inch
two instead of zero), what would I have to do to find the true
heights of everyone? If you think about it you will see that I must
subtract two inches from each measurement to get the true heights
(because the start position was too high). This example of a linear
transformation is one in which we simply shift the numbers up on
the same scale. Notice that even though all the numbers move, the
relationship between values is not affected. X X + 2 55 57 57 59 58
510 510 60 You will need to know how linear transformations affect
the mean and standard deviation of a distribution as well. How does
adding or subtracting a constant affect the mean and standard
deviation? How does multiplying and dividing a constant affect the
mean and standard deviation?
-
When adding or subtracting a constant from a distribution, the
mean will change by the same amount as the constant. The standard
deviation will remain unchanged. This fact is true because, again,
we are just shifting the distribution up or down the scale. We do
not affect the distance between values. In the following example,
we add a constant and see the changes to the mean and standard
deviation. X X +5 1 6 2 7 3 8 4 9 5 10 = 3 = 8 = 1.41 = 1.41 The
effect is a little different when we multiply or divide by a
constant. For these transformations the mean will change by the
same amount as the constant, but this time the standard deviation
will change too. That is because when we multiply numbers together,
for example, we change the distance between values rather than just
shifting them up or down the scale. In the following example, we
multiply a constant and see the changes to the mean and standard
deviation. X X * 5 1 5 2 10 3 15 4 20 5 25 = 3 = 15 = 1.41 = 7.91
Z-Transformation The z-transformation is a linear transformation,
just like those we have discussed. Transforming a raw score to a
Z-score will yield a number that expresses exactly how many
deviations from the mean a score lays. Here is the formula for
transforming a raw
score in a population to a Z-score: z = X
Notice that the distance a score lies from the mean is now
relative to how much scores deviate in general from the mean in the
population. Regardless of what the raw score values are in the
population, when we use the Z-transformation we obtain a standard
measure of the distance of the score from the mean. Anytime Z=1,
the raw score that produced the Z is exactly one standard deviation
from the mean for that population. Anytime Z=1.5, the raw score
that produced the Z is exactly 1.5 standard deviations from the
mean for that population.
-
Think for a minute about what it means to know how many standard
deviations from a mean a score lays. Consider our simple
distribution example. X 1 2 3 4 5 = 3 = 1.41 What z-score will we
expect the value 3 to have in this example? That is, how many
standard deviations from the mean is 3? The answer is that it is at
the mean, so it is zero standard deviations from the mean and we
will get a z-score of zero for the original value of three. Now
consider the value 1 in the distribution. What z-score will we
expect to get for this score? Will it be less than one standard
deviation or more than one standard deviation away from the mean?
You can estimate the z-score by counting from the mean. One
standard deviation is 1.41 units. Counting down from the mean the
value 2 is one unit from the mean. Thats a little less than one
standard deviation. We have to go down 1.41 units from the mean
before we reach one standard deviation. So, when we get to 1 on the
scale, we are two units from the mean and a little more than one
standard deviation below the mean. Lets transform the simple
distribution into a distribution of z-scores by plugging each value
into the z-formula: X Z-Tranformation Z
1 ==41.1
31z -1.42
2 z = X = -.71
3 z = X = 0
4 z = X = .71
5 z = X = 1.42
= 3 = 0 = 1.41 = 1
-
The value of 1 is 1.42 standard deviations below the mean. The
value 2 is .71 standard deviations below the mean, and so on.
Notice that the mean is at zero, so any scores below the mean in
the original distribution will always have a negative z-score and
any score above the mean will have a positive z-score. Properties
of the z-distribution. Also notice in the above example that we had
to compute the mean and standard deviation of the simple
x-distribution in order to compute the z-score. We can compute the
mean and standard deviation of the resulting z-distribution as
well. The mean of the z-distribution will always be zero, and the
standard deviation will always be one. These facts make sense
because the mean is always zero standard deviations away from the
mean, and units on the z-distribution are themselves standard
deviations. Using Z-scores to Compare Values Since z-scores reflect
how far a score is from the mean they are a good way to standardize
scores. We can take any distribution and express all the values as
z-scores (distances from the mean). So, no matter scale we
originally used to measure the variable, it will be expressed in a
standard form. This standard form can be used to convert different
scales to the same scale so that direct comparison of values from
the two different distributions can be directly compared. For
example, say I measure stress in the average college sophomore on a
scale between 0 and 30 and find the mean is 15. Another researcher
measures stress with the same population, but uses a scale that
ranges from 0 to 300 with a mean of 150. Who has more stress, a
person with a stress score of 15 from the first distribution or a
person with a stress score of 150 from the second group? The value
of 150 is a much larger number than 15, but both are at the mean of
their own distribution. Thus, they will both have the same z-score
of zero. Both values indicate an average amount of stress.
Consider another example. Lets say that Joan got an x = 88 in a
class that had a mean score of 72 with a standard deviation of 10 (
= 72, = 10).
-
In a different class lets say Bob got a x = 92. The mean for
Bobs class, however, was 87 with a standard deviation of 5 ( = 87,
= 5). Who had the higher grade relative to their class? If you
think about it for a second you will know that Joans score of 88 is
much higher relative to the average of 72 compared to Bobs score of
92 to the average of 87. We can easily compare the values, however,
if we simply compute the z-score for each. Joan
6.1
1016
107288 ===Z
Joan has the higher score because she is 1.6 standard deviations
above the mean, and Bobs score is only 1 standard deviation above
the mean. Bob
1
55
58792 ===Z
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Lesson 7 Z-Scores and Probability
Outline Introduction Areas Under the Normal Curve Using the
Z-table Converting Z-score to area -area less than z/area greater
than z/area between two z-values Converting raw score to Z-score to
area Converting area to Z-score to raw score Introduction/Area
Under the Curve Please note that area, proportion and probability
are represented the same way (as a decimal value). Some examples
require that you convert the decimal value to a percentage. Just
move the decimal place to the right two places to turn the decimal
into a percentage. Start this section by reviewing the first two
topics in the above outline on the web page. Find the Z-score
animated demonstrations or click here
http://faculty.uncfsu.edu/dwallace/sz-score.html Using the Z-table
Knowing the number of standard deviations from the mean gives us a
reliable way to know how likely a score is for a population. There
is a table of z-scores that gives the corresponding areas or
probabilities under the curve. You will need the z-table in
Appendix B of your text for this discussion. See page A-24 through
A-26 in your text. The table shows the z-score in the left column,
and then the proportion or area under the curve in the body, and
finally there is a third column that shows the proportion or area
under the curve in the tail of the distribution. Whenever we
compute a z-score it will fall on the distribution at some point.
The larger portion is the body, and the smaller portion is the
tail.
-
If the z-score is positive, then the body will be the area that
is below that z-score, and the tail will be the area that is above
that z-score. If the z-score is negative, then the body will be the
area that is above that z-score, and the tail will be the area that
is below that z-score.
Converting a Z-Score to an Area Finding areas below a z-score
What area lies below a z-score of +1? If we look this z-score up on
the table we find that the area in the body is .8413 and the area
in the tail is.1587. Since the z-score is positive and we want the
area below the z-score, then we will want to look at the body area.
So, .843 is the proportion in the population that have a z = 1 or
less.
Now lets consider the situation if the z-score is negative. What
area lies below a z-score of -1.5? You will not find negative
values on the table. The distribution is symmetric, so if we want
to know the area for a negative value we just look up the positive
z-score to find the area. We will use a different column on the
table, and that is why we must consider whether z is positive or
negative when using the table. If we look this z-score (1.5) up on
the table we find that the area in the body is .9332, and the area
in the tail is .0668. Since the z-score is negative and we want the
area below that point we will be using the tail area. So, .0668 is
the proportion in the population below a z-score of -1.5.
-
Finding areas above a z-score The process for this type of
problem is the same as what we have already learned. The only
difference is in which column we will be using to answer the
question. What area lies above a z-score of +1? If we look this
z-score up on the table we find that the area in the body is .8413
and the area in the tail is.1587. Since the z-score is positive and
we want the area above the z-score, then we will want to look at
the tail area. So, .1587 is the proportion in the population that
have a z = 1 or more. Now lets consider the situation if the
z-score is negative. What area lies above a z-score of -1.5? Again,
you will need to look up the positive z-value for 1.5. If we look
this z-score up on the table we find that the area in the body is
.9332, and the area in the tail is .0668. Since the z-score is
negative and we want the area above that point we will be using the
body area. So, .9332 is the proportion in the population above a
z-score of -1.5. Finding areas between two z-scores When we have
two different z-scores and want to find the area between them, we
first must consider if both values are on the same side of the
mean, or if one value is positive and the other negative. For our
table, if the values are either both positive zs or both negative
zs, we can find the tail area for both z-scores and subtract the
two areas. You could just as easily find the two body areas for
both z-scores and subtract them as well. For example, what is the
area between Z = 1 and Z = 1.5? Since both scores are positive and
we want the area between them, we will look up the tail area and
subtract the two table values. Note that you never subtract
z-scores, only areas from the table.
-
On the other hand, if you have one positive and one negative
z-score then you must use the body area for either one of the
z-scores, and the tail area for the other. Once you get the two
areas off the table, then you subtract the two areas. For example,
what is the area between Z = -1 and Z = 1.5? Since one score is
positive and the other negative, the area we are looking for will
cross the mean. Use the body area for one value and the tail area
for the other. Once you get these values off the table, subtract
them to find the area in between.
Converting a Raw Score to a Z-score and then into an Area These
problems are exactly the same as the others we have been working.
You must still find areas above/below/and between two z-scores, but
now you must first compute the z-value using the z-formula before
using the table. For example lets look at IQ scores for the
population with a mean of 100 and standard deviation of 15 ( = 100,
= 15). What proportion of the population will have an IQ of 115 or
less? I first must compute Z. It is equal to z = 1. Now the
question becomes what proportion of scores lie above z = 1?
-
Please review other examples of this type of problem on the
web-page. Find the link to Z-scores and probability or click here
http://faculty.uncfsu.edu/dwallace/sz-score2.html Converting an
Area to a Z-score and then into a Raw Score For these problems we
will be doing the same process we have been doing, but everything
will be done in reverse order. We will start with a given area or
proportion. You then use the z-table to find the area. However,
when you use the z-table for these problems you must look up the
area in either the body or tail column and then trace it back to
find the z-score. Once we get the z-score we will plug in the
values we know and solve for X in the z-score equation. For
example, IQ scores for the population with a mean of 100 and
standard deviation of 15 ( = 100, = 15), what score cuts off the
top 10% of the distribution? Notice that these questions are always
asking what the score is for a certain point. We are solving for X
now. Prior examples were all asking for an area or proportion. The
first step to solving this type of problem is to find the Z-score.
We wont be computing z, but instead finding it from the table.
Since we want the top 10%, we will be looking for the area on the
table where the tail is .10 and the body is .90. You can look in
either column, and it might help to draw the distribution in order
to be sure you are using the right column. Make sure you are not
using the z-column at this point.
We find the z-score that leaves the tail at the top of the
distribution equal to .10 is Z = 1.28. Always use the z-score
closest to the area of interest. Once we have that number,
-
we can plug in what we know into the z-formula and solve for X.
Alternatively, if you have trouble with algebra, you can use the
following formula: X = Z +
Special note for values in the lower 50% of the distribution:
Whenever we want to find a z-score for a value below the mean, we
must remember to make the value negative. Recall that the z-table
only gives positive z-values. If the value is below the mean, then
you must remember to insert the negative sign before doing the
computation. For example, if I were looking for the IQ score for
the bottom 10% of the distribution, in the above example, then I
would have look up a tail area of .10 or body area of .90. The
z-score we need is -1.28 even though the table shows only the
positive value. Please review other examples of this type of
problem on the web-page. Find the link to Z-scores and probability
or click here http://faculty.uncfsu.edu/dwallace/sz-score2.html
-
Lesson 8 Probability
Outline Probability of an Event Probability of Single Events
Probability of Multiple Events -without replacement -mutually
exclusive events Conditional Probability Probability of an Event
There are three classes of events: Impossible
Events--------------Possible Events-----------Certain Events P = 0
P = 0 to 1 P = 1 The probability (P) of an impossible event is zero
because there is zero chance of it happening. A certain event has a
probability of one or 100% because it will always happen. Most of
the events we will be interested in are possible events. These
probabilities will always have a value between zero and one. Always
leave your answer in decimal form, instead of fractional form.
Simply divide out any fraction you have by dividing the top number
by the bottom number. So, is 0.25. A simple experiment we could run
to examine probabilities is to roll a six-sided die. What is the
probability of rolling a 5 on a single die roll? Most people know
it is 1/6 or .167 because there is only one side that is a 5 and
there are six sides on the die. You can determine the probability
of any even in this manner.
Probability itemsTotalcriteriainitemsP
__#___#)( =
So, we were only looking for one side in the last problem, out
of a total of six sides. Another experiment we could use to look at
probabilities is drawing cards from a standard deck. If I draw a
card from a deck of cards what is the probability it is a
heart?
P 25.041
5213)( ===Heart
Note that a standard deck of cards has 52 cards with 13
hearts/13 clubs/13 diamonds/13 spades. Since diamonds and hearts
are red cards and the rest are black, there are 26 red and 26 black
cards.
-
Probability of Single Events An individual event is a single
event. With single events we are measuring the likelihood of a one
thing happening. We might be interested in different outcomes, but
we are still just going to roll the die once or draw a single card
from a deck. For example, what is the probability that I roll a 5
or a 6 on a single die roll or P (5 or 6)? With single events you
will see this or connector and you will add the two individual
probabilities. So: P (5 and 6) = 1/6 + 1/6 = .167 + .167 = .334
What is the probability I draw a Heart or Club with a single draw
from a standard deck of cards? P (Heart or Club) P(Heart or Club) =
P(Heart) + P(Club) = 13/52 + 13/52 = .25 + .25 = .5 Probability of
Multiple Events With multiple events we will be interested more
than one outcome be realized. So, we will roll the die more than
once or draw more than one card from a deck. For example, what is
the probability of rolling a 5 and a 6 on two die rolls. To get
both a five and a six I will have to roll the die more than once.
When you see this and connector you will multiply individual
probabilities. P (5 and 6) = P(5) * P(6) = 1/6 * 1/6 = .167 * .167
= .028 We will only be dealing with independent events in this
section, or events that do not affect the outcome of other events.
With the card experiment, then, we will not look at multiple draws
where one draw could affect the probability of a separate event.
For example, what is the probability of drawing a Heart and a Club
from standard deck? P(Heart and Club) = P(Heart) * P(Club) = 13/52
* 13/52 = .25 * .25 = .062
-
Without Replacement Although we will focus on independent events
like the last example, we will also consider what happens to
probabilities in situations in which there is no replacement. The
above examples assumed that once we drew a card from the deck that
it was replaced before another draw was made. Notice that when
figuring how many total events there were we used 52 every time
because we assumed each draw was from a fresh deck. If the problem,
however, specifies that there is no replacement then we must take
this into account when figuring the probabilities. For example,
what is the probability of drawing a Heart and a Club from a deck
without replacement? When we count how many cards are left for the
Club draw, there will be one less card in the deck because we
already had to draw the Heart from the deck. Thus: P(Heart and
Club) = P (Heart) * P (Club) = 13/52 * 13/51 = .25 * .255 = .064 We
might also have to subtract a value from the numerator as well as
the denominator. Try to find the probability of drawing three red
cards from a deck without replacement. (Answer: 0.1176) Mutually
Exclusive Events Mutually exclusive events are events that cannot
happen together. For example, being a freshman and a sophomore are
mutually exclusive. You are either one or the other but not both.
For mutually exclusive events the probability the two events will
occur together must always equal zero. Conditional Probability With
conditional probabilities we will consider the probability of an
event given that some other event has already happened. Thus, these
are not independent events, and the rules we learned above will not
apply. For these problems frequency data (or counts) will be given
in a contingency table. This table will display the frequencies for
different combinations of events. For example, consider the
probability of having a computer or not, and living the U.S. or
elsewhere. In U.S. Not in U.S. Computer 30 15 No Computer 10 20
-
Before we consider conditional probabilities, lets look at some
of the types of questions we have already examined. For example,
what is the probability that choosing someone from our sample will
yield a person with a computer? To answer this question we will
need to add up the total for each row and column in the table: In
U.S. Not in U.S. Total Computer 30 15 45 No Computer 10 20 30 total
40 35 Since there are a total of 45 people in our sample with a
computer out of 75 total people, there is a 0.6 probability that a
random draw will yield a person with a computer. Now find the
probability that a random draw will yield someone with a computer
that is living in the U.S. Instead of looking in the total column
for this type of problem, we will use one of the original values.
There are 30 people living in the U.S. that also have a computer.
So, 30 out of the total of 75 people or 0.4 live in the U.S. and
have a computer. For a single event in a table like this one, use
the values in the margin or the totals, and divide by the total
number in the sample. For a combined event, use the original table
values out of the total. For conditional probabilities we will
restrict our sample to those items given to have already have
happened. For example we might know the probability of having a
computer and living in the U.S. for the entire sample. We could
make this conditional by saying what is the probability of having a
computer given that we know the person is living in the U.S.? With
the second question we are not asking the probability of picking a
person at random from the total, but instead we are restricting our
sample to just those that live in the U.S. For conditional
probabilities the total or denominator is the value given. For this
example it is the total for those living in the U.S. or 40. We want
to know what proportion have a computer out of these 40 people
living in the U.S. Since 30 of those in the U.S. have a computer
out of 40 the probability is 30/40 = .75 In this same example what
is the probability someone does not live in the U.S. given they
have a computer? We can write: P (not in U.S. | computer) Where the
first probability is the one we are interested in, the vertical
line means given and the second probability computer is what is
given. Again, our new total is those with a computer or 45. We want
to know the proportion out of these that are not in the U.S. or 15.
So, the conditional probability is 15/45 = .33
-
Lesson 9 Hypothesis Testing
Outline Logic for Hypothesis Testing Critical Value Alpha ()
-level .05 -level .01 One-Tail versus Two-Tail Tests -critical
values for both alpha levels Logic for Hypothesis Testing Anytime
we want to make comparative statements, such as saying one
treatment is better than another, we do it through hypothesis
testing. Hypothesis testing begins the section of the course
concerned with inferential statistics. Recall that inferential
statistics is the branch of statistics in which we make inferences
about populations from samples. Up to this point we have been
mainly concerned with describing our distributions using
descriptive statistics. Hypothesis testing is all about
populations. Although we will start using just one value from the
population and eventually a sample of values in order to test
hypotheses, keep in mind that we will be inferring that what we
observe with our sample is true for our population. We will want to
see if a value or sample comes from a known population. That is, if
I were to give a new cancer treatment to a group of patients, I
would want to know if their survival rate, for example, was
different than the survival rate of those who do not receive the
new treatment. What we are testing then is whether the sample
patients who receive the new treatment come from the population we
already know about (cancer patients without the treatment). Again,
even though we are talking about a sample, we infer that the sample
is just part of an entire population. The population is either the
one we already know about, or some new population (created by the
new treatment in this example). Logic 1) To determine if a value is
from a known population, start by converting the value to a z-score
and find out how likely the score is for the known population. 2)
If the value is likely for the known population then it is likely
that it comes from the known population (the treatment had no
effect). 3) If the value is unlikely for the known population then
it is probably does not come from the population we know about, but
instead comes from some other unknown population (the treatment had
an effect).
-
4) A value is unlikely if it is less than 5% likely for the
known population. Any value that occurs 5% or more of the time for
the known population is likely and part of the known population.
The 5% cut-off point is rather arbitrary, and it will change as we
progress. For now we will use it as a starting point to illustrate
several concepts.
Lets look at a simple example. Say the earth has been invaded by
aliens that look just like humans. The only way to tell them apart
from humans is to give them an IQ test since they are quite a bit
smarter than the average human. Lets say the average human IQ (the
known population) is = 100 = 15. We want to know if Bob is an
alien. Bob has an IQ score of 115? Is it likely that he comes from
the known population and is human, or does he come from a different
alien population? To answer the question, first compute the
z-score.
11515
15100115 ===Z Next, find out how likely this z-score is for the
population.
For hypothesis testing we will always be interested in whether
the value is extreme for the population, or unlikely. Thus, we will
be looking in the Tail Column when deciding if the value is
unlikely.
-
So, the likelihood of observing an IQ of 115 or more is .1587.
Since the probability is not less than 5% or .05 we have to assume
Bob comes from the general population of humans. Say Neil has an IQ
of X = 30. Is it likely Neil comes from the general population?
Again we first compute the z-score, and then find how likely it is
to get that value or one more extreme.
21530
15100130 ===Z Next, find out how likely this z-score is for the
population.
So, the likelihood of observing an IQ of 130 or more is .0228.
Since the probability is less than 5% or .05 we have to assume Neil
comes from a different population than the one we know about (the
general population of humans). Critical Value Critical Values are a
way to save time with hypothesis testing. We dont really have to
look up the probability of getting a particular value in order to
verify it is less than 5% likely. The reason for this fact is that
the z-score that marks the point where a value becomes unlikely
does not change on the z-scale. That is, there is only one z-score
at the that is 5% likely. Any z-score beyond that point is less
than 5% likely. Thus, I dont have to look up each particular area
when I compute my z-score. Instead I only have to verify that the
z-score I computed is more extreme than the one that is 5% likely.
So, we can stop once we compute the z-score without reference to
the z-table. What z-score will be exactly 5% likely for any
population? This is the z-score we will make comparisons against.
Use the z-table to determine the z-score that cuts off the top 5%
of scores. Finding the critical value is important, and will be one
of the steps that must be performed anytime we conduct a hypothesis
test. Since we will be doing hypothesis testing from this point on,
many points on subsequent exams will come from just knowing the
critical value.
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So, Z = 1.64 is the critical value. Any value more extreme than
1.64 is unlikely, and all other events will be likely for the known
population. If Z=2.1, or Z=1.88 you conclude that the value is
unlikely, and so must be part of a different population. If Z =
1.5, or Z =0.43 you conclude that the value is likely, and so must
be part of the known population. Note that we were only working on
one side of the distribution in the above problem. If we were
interested in a value that was below the mean, instead of above it,
then we would flip our decision line to the other side. Since the
distribution is symmetric, the numbers will not change.
Alpha Alpha is the probability level we set before we say a
value is unlikely for a known population. The critical value we
just found is only one that we will use. It assumes that a value
must be less than 5% likely to be unlikely, and therefore part of a
different population. Alpha was .05 ( = .05) for that example.
Sometimes researchers want to be very sure before they decide a
value is different. Thus, we will also use an alpha level of .01 or
1% as well. If alpha is .01, then a value must be less than 1%
likely before it is said to be unlikely for a known population. If
alpha is 1% then the critical value will be different than the one
we found above. Alpha is given in every problem, but you must use
that information to determine the critical value. What z-score will
be exactly 1% likely for any population? This is the z-score we
will make comparisons against when alpha is set to 1% ( = .01). Use
the z-table to determine the z-score that cuts off the top 1% of
scores just like the last example. Use the Tail column and find
.01. You could also look in the Body Column and find .99. We
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will use Z = 2.33 when Alpha is set to the 1% level. Also note
that when we are interested in determining if values below the mean
are unlikely our critical value will be negative. One-Tail versus
Two-Tail Tests Another factor that will affect our critical value
is whether we are performing a one or a two-tail test. The critical
values we have looked at so far were for one-tail tests because we
were only looking at one tail of the distribution at a time (either
on the positive side above the mean or the negative side below the
mean). With two-tail tests we will look for unlikely events on both
sides of the mean (above and below) at the same time. I will
discuss how to determine if a problem is a one or a two tail test
in a later lesson, but lets go ahead and find the critical values
for two-tailed test the same way we did the one-tail tests above.
Lets begin with an alpha level of 5%. We still want 5% of our
events to be unlikely and 95% of our events to be likely for the
known population. Now, however, we want to be looking for unlikely
events in both directions at the same time. So, we will split the
unlikely block into two parts, each half the total 5% area.
What z-scores will then mark the middle 95% of our distribution?
You will have to look up an area of .025 in the Tail Column of the
z-table.
The process for finding the two-tail critical values when alpha
is set to .01 is the same. This time we will want 99% of our values
in the middle, leaving only .005 or half of one percent on each
side. Can you find the critical value on the z-table (answer: z =
2.58).
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So, we have learned four critical values. 1-tail 2-tail = .05
1.64 1.96/-1.96 = .01 2.33 2.58/-2.58 Notice that you have two
critical values for a 2-tail test, both positive and negative. You
will have only one critical value for a one-tail test (which could
be negative).
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Lesson 10 Steps in Hypothesis Testing
Outline Writing Hypotheses -research (H1) -null (H0) -in symbols
Steps in Hypothesis Testing -step1: write the hypotheses -step2:
find critical value -step3: conduct the test -step4: make a
decision about the null -step5: write a conclusion Writing
Hypotheses Before we can start testing hypotheses, we must first
write the hypotheses in a formal way. We will be writing two
hypotheses: the research (H1) and the null (H0) hypothesis. The
research hypothesis matches what the researcher is trying to show
is true in the problem. The null is a competing hypothesis.
Although we would like to directly test the research hypothesis, we
actually test the null. If we disprove the null, then we indirectly
support the research hypotheses since it competes directly with the
null. We will discuss this fact in more detail later in the lesson.
Again, the research hypothesis matches the research question in the
problem. Lets take a look at a sample problem: Suppose some species
of plants grows at 2.3 cm per week with a standard deviation of 0.3
( = 2.3 = 0.3). I take a sample plant and genetically alter it to
grow faster. The new plant grows at 3.2 cm per week (X = 3.2). Did
the genetic alteration cause the plant to grow faster than the
general population? Set alpha = .05. Lets focus on writing
hypotheses, rather than any other steps we have learned for now. In
order to write the research hypothesis look at what the researcher
is trying to prove. Here we are trying to show that the genetically
altered plant grows at a faster rate than unaltered plants. Thats
what we want the research hypothesis to say. However, when you
write your hypotheses, be sure to include three elements: 1)
explicitly state the populations you wish to compare. For now, one
will be a treatment population and the other will always be the
general population. 2) State the dependent variable. We have to be
explicit about the scale on which we expect to find differences. 3)
State the type or direction of the effect. Are we predicting the
treatment population will be greater or less than the general
population (1-tail)? Or, are we looking for differences in either
direction at the same time (2-tail)? The above problem is one-tail
since we are looking for a growth rate higher than the average.
Look for words that indicate a direction in the problem for
one-tail test (e.g. higher/lower, more/less,
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better/worse). It would be two-tailed if the problem had stated
that we expected a different growth rate than the general
population. Different could be higher or it could be different
because it is lower. The current example is easy to translate into
a hypothesis, but check the homework packet because the wording is
not always so obvious. For the research hypotheses (denoted by H1
): H1: The population of genetically altered plants grows faster
than the general population. You could vary the wording a bit, as
long as you include the three elements. Notice that we state both
the treatment population and the population we will compare that
to, the general population. Growth rate is the dependent variable,
and we indicate the direction by saying it will grow faster. The
null hypothesis (denoted by H0) is a competing hypothesis. Its
basically the opposite of the research hypothesis. In general it
states that there is not effect for our treatment or no differences
in our populations. For this example: H0: The population of
genetically altered plants grows at the same or lower rate as the
general population. Ive included the same or lower wording for the
one-tail test because we want to cover all the possible outcomes of
the test. We only want to show that the treatment population grows
faster. If they end up growing slower it wont support the research
hypothesis, so we include left-over elements with the null. For
two-tail tests, substitute different for the word faster in the
research hypothesis. The two-tail null would say the groups are do
not differ. In Symbols We can also write the hypothesis in
notational form. We will restate both the null and research
hypotheses in symbols we have been using for our formulas. Thus:
H1: gen.alt. > 2.3 H0: gen.alt < 2.3 Notice that we represent
the treatment population with a mu (). We do this because we want
to make inferences about the population, not the single value
sample I am using to test the hypothesis. Our inferences will be
that the entire population the plant comes from grows at a faster
rate. The value of 2.3 is the general population mean we are
comparing against. Although it is represented with a mu in the
problem, we dont the
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symbol because we know the exact value for that population. For
two-tail test we simply change the direction arrows to
equal/not-equal signs (an = sign for the null and / sign for the
research hypothesis). Steps in Hypothesis Testing Now we can put
what we have learned together to complete a hypothesis test. The
steps will remain the same for each subsequent statistic we learn,
so it is important to understand how one step follows from another
now. Lets continue with the example we have already started:
Suppose some species of plants grows at 2.3 cm per week with a
standard deviation of 0.3 ( = 2.3 = 0.3). I take a sample plant and
genetically alter it to grow faster. The new plant grows at 3.2 cm
per week (X = 3.2). Did the genetic alteration cause the plant to
grow faster than the general population? Set alpha = .05. Step 1:
Write the hypotheses in words and symbols H1: The population of
genetically altered plants grows faster than the general
population. H0: The population of genetically altered plants grows
at the same or lower rate as the general population. H1: gen.alt.
> 2.3 H0: gen.alt < 2.3 Step 2: Find the critical value for
the test Since alpha is .05, and it is a one-tail test because we
think our treatment will produce plants that grow faster than the
general population: Zcritical =1.64 Step 3: Run the test Here we
find out how likely the value is by computing the z-score.
33.09.0
3.03.22.3 ===Z
Step 4: Make a decision about the Null Reject the Null or Fail
to Reject the Null (retain the null) are the only two possible
answers here. Since the value we computed for the z-test is more
extreme than the
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critical value, we reject the Null. Graphically, though not
required for the answer, we have:
Note that we are testing the Null. It is either proven or
disproven. We never prove the research hypothesis, even thought
that is our intent. Instead, if we disprove the null, we indirectly
support the research hypothesis. This is true, because you will
notice that our decision is based on the statistical test that the
treatment value is not likely to have come from the same
population. We infer it is a different population, but we actually
prove that it is not from the same population. It may seem like a
matter of semantics, but indulge me on this one. Step 5: Write a
conclusion For this example, we conclude: The population of
genetically altered plants grows at a different rate than the
general population. Although we have a conclusion in step 4, write
a conclusion here in plain language without any statistical jargon.
What did our test show? If you reject the null, then the then there
was a difference (treatment had an effect). The research hypothesis
is your conclusion (you can simply restate it from Step 1). If you
fail to reject the null, then the null hypothesis is your
conclusion (again, you can just rewrite it from Step 1).
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Lesson 11 Hypothesis Testing with a Sample of Values
We have looked at the basics of hypothesis testing using the
z-formula we had already learned. However, we never test a
hypothesis based on one individual from a population. Instead, we
will want to have a sample of values to test against the
population. The formula we will want to use has a minor change from
the one we have been using.
x
Xz = , where
x = n
Notice that there is sample mean now in the numerator instead of
just a single x-value. Often this will be given just like the
x-value in prior problems, but now you may also have to compute it
from the sample. Compute x first for the denominator by dividing
the standard deviation by the square root of the given sample size
(n). Once you get that number plug it in as the denominator in the
z-score formula. The rest of this lesson is devoted to the theory
behind the changes we make when moving from tests with a single
x-value to tests with samples of x-values. There are no
computational additions for the exam other than the formula change
above. However, you should be concerned with understanding the
conceptual meaning of this lesson. At a minimum you should be able
to recognize the rules of the Central Limit Theorem for the exam
(detailed below).
The lesson continues on the web page. Take notes on the sampling
distribution page, the standard error page, and the standard error
with hypothesis testing page. Links to these pages are provided
below. It is important to review each one. Again, these lessons
contain conceptual information for the most part, however, the last
page is devoted to the computations you will perform for the
exams.
Sampling Distributions
Standard Error
Standard Error and Z-score Hypothesis Testing
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Lesson 12 Errors in Hypothesis Testing
Outline Type I error Type II error Power Examples in the real
world Anytime we make a decision about the null it is based on a
probability. Recall that we reject the null when it tests a value
that is unlikely for the known population. Extreme values are
unlikely for the population, but not impossible. So, there is
always some chance that our decision is in error. Note that we will
never know whether we know we have made an error or not with our
hypothesis test. When running a test, I only know what my decision
is about the test, and not the true state of reality. Thus, this
discussion on errors is strictly theoretical. Type I Errors
Whenever a value is less than 5% likely for the known population,
we reject the null, and say the value comes from some other
population.
Notice that we are saying the value is really from another
population distribution out there that we dont know about.
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However, some of the time the value really does come from the
known population. Notice that even though the value represented is
beyond the critical value it still lies under the curve for the
normal population. We reject any values in this range, even though
they really are part of the known population. When we reject the
null, but the value really does come from the known population a
Type I error has been committed. A Type I error, then, happens when
we reject the null when we really should have retained it. Note
that a Type I error can only occur when we reject the null. The
part of the distribution that remains under the curve for the known
population but is beyond our critical value in the region of
rejections is alpha (). When we set alpha we are setting the
probability of making a Type I error.
Type II Errors Whenever a value is more than 5% likely for the
known population, we retain the null, and say the value comes from
the known population. But, some of the time the value really does
come from a different unknown population.
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Notice in this situation the value is below the critical value,
so we retain the null. However, the value is still under the
unknown population distribution, and may in fact come from the
unknown population instead. Thus, when I retain the null, when I
should really have rejected it I commit a Type II error. The
probability of making a Type II error is equal to beta and not
strictly defined by alpha. Although we know the probability of a
Type I error because we set alpha, a Type II error takes in a few
more factors than that. You can see the region of Beta () below.
Notice that it is the area below the critical value, but that is
still part of the other unknown distribution.
Power Power is the probability of correctly rejecting the null
hypothesis. That is, it is the probability of rejecting the null
when it is really false. Again, we never really know if the null is
false or not in reality. Power is another way of talking about Type
II errors. Such errors have been recognized as a problem in the
behavioral sciences, so it is important to be aware of such
concepts. However, we will not be computing power in this course.
You can ignore the power demonstration on the web page for that
reason. An easy way to remember all these concepts might be to put
them in a table, much like your textbook does.
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Examples of Errors in the Real World Another way to think about
Type I and Type II errors is to think of them in terms of false
positives and false negatives. A Type I error is a false positive,
and a Type II error is a false negative. A false positive is when a
test is performed and shows an effect, when in fact there is none.
For example, if a doctor told you that you were pregnant, but you
were not then it would be a false positive result. The test shows a
positive result (what you looking for is there), but the test if
false. A false negative is when a test is performed and shows no
effect, when in fact there is an effect. The opposite situation of
the above example would apply. A doctor tells you that you are not
pregnant, when if fact you are pregnant. The test shows a negative
result (what you are looking for is not there), but the test is
false. Lets look at another example. A sober man fails a blood
alcohol test. What type of error has been committed (if any)? For
this type of problem you will get two pieces of information. First,
whether the test was positive or negative. The test is positive if
what you are looking for is found. It is negative if the test shows
what you are looking for is not there. The second piece of
information is whether the test is in error or not (false or true
test). Thus, for this example, the test is positive because if you
fail a blood alcohol test it is showing that there is alcohol in
your system. You are positive for alcohol in that case. Since the
man is sober, the test is false. So, here we have a false positive
test or Type I error.
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Lesson 13 Hypothesis Testing with the t-test Statistic
Outline Unknown Population Values The t-distribution -t-table
Confidence Intervals Unknown Population Values When we are testing
a hypothesis we usually dont know parameters from the population.
That is, we dont know the mean and standard deviation of an entire
population most of the time. So, the t-test is exactly like the
z-test computationally, but instead of using the standard deviation
from the population we use the standard deviation from the
sample.
The formula is: t = X
sx , where sx = sn
The standard deviation from the sample (S), when used to
estimate a population in this way, is computed differently than the
standard deviation from the population. Recall that the sample
standard deviation is S and is computed with n-1 in the denominator
(see prior lesson). Most of the time you will be given this value,
but in the homework packet there are problems where you must
compute it yourself. The t-distribution There are several
conceptual differences when the statistic uses the standard
deviation from the sample instead of the population. When we use
the sample to estimate the population it will be much smaller than
the population. Because of this fact the distribution will not be
as regular or normal in shape. It will tend to be flatter and more
spread out than population distribution, and so are not as normal
in shape as a larger set of values would yield. In fact, the
t-distribution is a family of distributions (like the
z-distribution), that vary as a function of sample size. The larger
the sample size the more normal in shape the distribution will be.
Thus, the critical value that cuts off 5% of the distribution will
be different than on the z-score. Since the distribution is more
spread out, a higher value on the scale will be needed to cut off
just 5% of the distribution. The practical results of doing a
t-test is that 1) there is a difference in the formula notation,
and 2) the critical values will vary depending on the size of the
sample we are using. Thus, all the steps you have already learned
stay the same, but when you see that the problem gives the standard
deviation from the sample (S) instead of the population (), you
write the formula with t instead of z, and you use a different
table to find the critical value. The t-table Critical values for
the t-test will vary depending on the sample size we are using, and
as usual whether it is one-tail or two-tail, and due to the alpha
level. These critical values
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are in the Appendices in the back of your book. See page A27 in
your text. Notice that we have one and two-tail columns at the top
and degrees of freedom (df) down the side. Degrees of freedom are a
way of accounting for the sample size. For this test df = n 1.
Cross index the correct column with the degrees of freedom you
compute. Note that this is a table of critical values rather than a
table of areas like the z-table. Also note, that as n approaches
infinity, the t-distribution approaches the z-distribution. If you
look at the bottom row (at the infinity symbol) you will see all
the critical values for the z-test we learned on the last exam.
Confidence Intervals If we reject the null with our hypothesis
test, we can compute a confidence interval. Confidence intervals
are a way to estimate the parameters of the unknown population.
Since our decision to reject the null means that there are two
populations instead of just the one we know about, confidence
intervals give us an idea about the mean of the new unknown
population. See the Confidence Interval demonstration on the web
page or click here http://faculty.uncfsu.edu/dwallace/sci.html for
the rest of the lesson.
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Lesson 14 Independent Samples t-test
Outline No Population Values Changes in Hypotheses Changes if
Formula -standard error Pooled Standard Error -weighted averages
Critical Values -df Sample Problem No Population Values With the
independent samples t-test we finally reach the point where we have
no population values. This fact is important because when we test
hypotheses we are usually testing an idea and a population that we
know nothing about. Think about the kinds of scientific discoveries
you hear about often. New treatments for diseases, new drugs, or
new techniques for improving depression all involve testing a
population created by the treatment or drug or technique. So, with
the independent samples t-test we will compare two sample values
directly. Note that we are still making the inference about the
populations from which the samples are drawn. Changes in Hypotheses
All hypotheses from this point on in the course will be two-tailed.
In addition, since we no longer no any population values we will
use mu to represent both populations. So for example, H0: diet =
placebo H1: diet placebo Formula Changes
Recall the formula for the t-test we have been using: t = X
sx , where sx = sn
The numerator will now have two sample values )( 21 XX instead
of one sample and one population. The denominator, recall, is the
standard error (the standard deviation divided by the square root
of the sample size).
Our standard error (denominator) was: sx = sn Remember that the
standard error measures variability we expect to see among samples.
Now that we have two samples we will want to include the estimate
of variability fro