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Chapter 10 - 12 – Significance/Hypothesis Testing Lesson
Objectives: In this chapter you will learn how to estimate the true
proportion/mean of an entire population and how to test a claim
about the proportion/mean of an entire population. Remember, you
cannot gather information from every member of the population, so
you collect information about a portion of the population (you take
a sample). You could then use that sample to estimate the
population proportion/mean or to test a claim that was made about
the population proportion/mean.
Date Topics Objectives: Students will be able to:
Assignments
Mar 25
10.2 The Reasoning of Significance Tests, Stating Hypotheses,
Interpreting P-values, Statistical Significance
10.4 Type I and Type II Errors, Planning Studies: The Power of a
Statistical Test
• State correct hypotheses for a significance test about a
population proportion or mean.
• Interpret P-values in context. • Interpret a Type I error and
a Type II error
in context, and give the consequences of each.
• Understand the relationship between the significance level of
a test, P(Type II error), and power.
Read TPS Section 10.2 – 10.4
[Watch Chapter 11
Videos 1 – 4]
Mar 26
12.1 Carrying Out a Significance Test, The One-Sample z Test for
a Proportion
12.2 Two-Sided Tests, Why Confidence Intervals Give More
Information
• Check conditions for carrying out a test about a population
proportion.
• If conditions are met, conduct a significance test about a
population proportion.
• Use a confidence interval to draw a conclusion for a two-sided
test about a population proportion.
Read TPS Section 12.1 – 12.2,
[Watch Chapter 11
Videos 5 – 8]
Mar 27
11.1 Carrying Out a Significance Test for 𝜇, The One Sample t
Test, Two-Sided Tests and Confidence Intervals
11.2 Inference for Means: Paired Data, Using Tests Wisely
• Check conditions for carrying out a test about a population
mean.
• If conditions are met, conduct a one-sample t test about a
population mean 𝜇.
• Use a confidence interval to draw a conclusion for a two-sided
test about a population mean.
• Recognize paired data and use one-sample t procedures to
perform significance tests for such data.
Read TPS Section 11.1 – 11.2
[Watch Chapter 11
Videos 9 – 12]
Chapter 11 Packet Due April 5
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10.2 – Reasoning of Significance Tests & Stating Hypotheses
[VIDEO #1] The two most common types of formal statistical
inference are:
1. Confidence Intervals -
______________________________________________________________
2. Significance Tests -
_______________________________________________________________
Both types of inference are based on the sampling distribution
of statistics à Chapter 9 stuff!
In the following problems, determine if a Confidence Interval
should be constructed or a Hypothesis Test carried out.
1. You measure the heights of a random sample of 400 high school
sophomore males. The sample mean is 𝑥 = 66.2 inches. Suppose that
the heights of all high school sophomore males follow a normal
distribution with unknown mean µ and standard deviation σ = 4.1
inches. You desire to estimate the mean height for all high school
sophomore males.
2. A researcher wishes to determine if students are able to
complete a certain pencil-and-paper maze more quickly while
listening to classical music.
3. An engineer designs an improved light bulb. The previous
design had an average lifetime of 1200 hours. The mean lifetime of
a random sample of 2000 of the new bulbs is found to be 1251 hours.
Although the difference is small, he would like to know if the
difference is statistically significant.
******************************************************************************************************************************
Just like CI’s might not capture the population parameter of
interest (and we don’t know it), we might make a decision with a
significance test that we believe is the right decision but ends up
being the wrong one. Let’s make the claim p = 0.62 is the true
proportion of CHS students who do not drink alcohol. Many of you
believe that number is an overestimate, so you believe, in fact, p
< 0.62. Let’s say we take a SRS of 100 students and ask them if
they have drunk alcohol in the past 30 days. We find 𝑝 = 0.58 from
our survey. Two things could happen as a result:
1. The initial claim (p = 0.62) is true, but because of sampling
variability, we just happened to collect a sample proportion (𝑝 =
0.58) that is lower than the claim.
2. The initial claim (p = 0.62) is false, so the sample
proportion (𝑝 = 0.58) was not unlikely to happen; thus proving our
belief that, in fact, p < 0.62.
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Stating Hypotheses We will need to write {our claims} and
{beliefs about our claims} in a more mathematical way. The initial
claim you want to try to prove is wrong is the
__________________________________, or ______.
• _______ will ALWAYS use _______________________________!!! The
belief you want to prove is right is the
_____________________________________________, or ______
• _______ will ALWAYS use one of the three
_____________________________ ( ____ , ____ , ____ )
o If you use ______ or _______ for Ha, then it is called a
______________________ test.
o If you use ______ for Ha, then it is called a
________________________ test. Hypotheses ALWAYS refer to a
___________________________, never to a ____________________. In
addition to writing the two hypotheses, you MUST define the
___________________ of interest (or write what each hypothesis
means to the problem…this takes more writing/effort!). Therefore,
we would write our hypotheses using the previous example as: Ho:
__________ (
_________________________________________________________________ )
Ha: __________ (
_________________________________________________________________ )
_______ =
_________________________________________________________________________
Example: Zach Holland is an avid golfer who would like to improve
his play. A friend suggests getting new clubs and lets Zach try out
his 7-iron. Based on years of experience, Zach has established that
the mean distance that balls travel when hit with his old 7-iron is
µ = 175 yards with a standard deviation of σ = 15 yards. He is
hoping that this new club will make his shots with a 7-iron more
consistent (less variable), so he goes to the driving range and
hits 50 shots with the new 7-iron.
(a) Describe the parameter of interest in this setting.
(b) State appropriate hypotheses for performing a significance
test.
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10.3 – Interpreting P-values & Statistical Significance
[VIDEO #2] Consider our judicial system. Arrested people are
innocent until proven guilty. We could say the null hypothesis is
“defendant is innocent” vs. the alternate hypothesis of “defendant
is guilty”. Prosecutors must provide enough evidence against the
defendant’s innocence to the jury for them to reject the null
hypothesis in favor of the alternate hypothesis that the defendant
is guilty of the crime. Do juries ever make the wrong decision? Of
course! The two types of mistakes that juries could make will be
discussed in the next video. For a jury to decide the defendant is
guilty there needs to be a certain level of evidence for them to
make that conclusion. If not enough evidence is provided to prove
guilt, then the jury will fail to reject the defendant’s innocence.
THIS IS A VERY IMPORTANT CONCEPT!!! à The jury NEVER PROVES the
defendant is innocent, but rather they were not provided enough
evidence to conclude the defendant was guilty. The amount of
evidence needed to change the jury’s mind from innocent (failing to
reject Ho) to guilty (rejecting Ho) is referred to as the
________________________________, or _____ level. This value is
directly tied to the confidence level from confidence intervals
from last chapter!!! A 95% CI = 5% α-level. A 99% CI = 1% α-level.
Common α-levels are 1%, 5%, and 10%. The significance (α) value
must be set BEFORE performing the significance test; otherwise,
it’s cheating! In terms of a pole vaulter, if you know how high you
can jump already, then you can set the bar low enough so you will
always jump over it! If you set the bar first before you know how
high you can jump, then you may or may not clear the jump. This
last scenario is what we wish to do statistically…to be fair! The
total evidence provided to the jury can be thought of as the
_____________________. The definition of this term is “the
probability, computed assuming Ho is true, that the statistic would
take a value as extreme as or more extreme than the one actually
observed is”. So, if the p-value (or total evidence assuming the
defendant is innocent) is as extreme as or more extreme than the
significance or α-level (that tipping point from assuming innocence
to concluding guilty), we can reject Ho (innocence) and conclude Ha
(guilt). This happens when the p-value ______________________ the
α-level. If the p-value < α-level, we will say that the data are
______________________________________________ at the α-level. In
that case, we _________________ the null hypothesis, Ho, and
conclude that there IS convincing evidence in favor of the
alternative hypothesis, Ha. In other words…
“IF THE P-VALUE IS TOO LOW (< α-level), THE
________________________!!!”
Example: Remember our golfer, Zach Holland, from the last video?
When he took his data and performed a significance test at the 5%
significance (α) level, he calculated a p-value of 0.0224.
(a) Interpret the results in context.
(b) Do the data provide convincing evidence against the null
hypothesis? Explain.
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10.4 – Type I and II Errors [VIDEO #3]
• Tests of Significance assess the _____________ of evidence
against the _______ hypothesis. We
measure evidence by the ____________, which is a probability
computed under the assumptions that
_____ is true. The alternative hypothesis (the statement we seek
evidence ______) enters the test only to
help us see what outcomes count against the null hypothesis.
• If our result is significant at level α, we reject ______ in
favor of ______. Otherwise, we fail to reject H0.
Consider a large hospital that needs 50,000 baby needles of
diameter 2mm. The shipment is in the right box,
but the needles look too big. The inspector is not going to
measure every needle. He will randomly select a
sample and measure the diameters. Surely there will be a little
variability in the measurements. Based on the
sample outcome, the shipment will either be accepted or
rejected.
Type I and Type II Errors
• When we must make a decision based on our sample we have two
choices:
1. Reject H0: We think the shipment of needles does not meet the
standards
2. Fail to Reject H0: We think the shipment of needles meets
standards
• There are two types of incorrect decisions:
1. We can reject a good shipment of needles – Waste of time for
the manufacturer, and the hospital
still needs the needles so they must place another order
2. We can keep a bad shipment of needles – Injure the
patients
Truth About the Population H0 True Ha True
Decision Based on
the Sample
Reject H0 Type I Error Correct Decision
Fail to Reject H0
Correct Decision Type II Error
Type I and Type II Errors:
If we reject H0 (accept Ha) when in fact H0 is true, this is a
______________________.
If we fail to reject H0 (reject Ha) when in fact Ha is true,
this is a ___________________.
Type I and II Error Calculations:
1. Type I: Probability of Type I Error = ____ (more on this in
the next video)
2. Type II: Probability of Type II Error = _____ (won’t
calculate this in here)
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Type I and Type II Errors: Court Case
Truth About the Defendant Innocent Guilty
Jury’s Decision Based on
the Evidence
Reject Innocence
Type I Error “Innocent Found Guilty” (Bad for the
Individual)
Correct Decision
Accept Innocence Correct Decision
Type II Error “Guilty Found Innocent”
(Bad for Society)
*************************************************************************************************************************
A new medication approved by the FDA has been accused of having
harsh side effects for its patients. It was known that the
medication may affect a patient’s kidneys. In order for kidneys to
be considered “failing”, they must have lost 80% of their ability
to function, thus resulting in dialysis. The pharmaceutical company
who manufactures the drug claims that in their clinical studies,
patients only experienced a kidney loss rate of 50% and therefore
never required dialysis. A large number of patients have had to
start dialysis treatment after being on the medication. A neutral,
third-party company decides to settle the issue and takes a SRS of
500 patients currently on the drug to measure what percentage their
kidneys are functioning at. 1. State the hypotheses and define the
parameter of interest. 2. Describe Type I error and Type II error
in this situation.
3. Which type of error is more serious? Why? 4. What
significance (α) level do you think should be used in this
situation? Why?
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10.4 – Type II Error & Power [VIDEO #4] 1. Type I Error:
_____________________________________________________________________
The probability of Type I Error is just ____, the significance
level of the test.
2. Type II Error:
_____________________________________________________________________
Type II Error occurs when you fail to reject the null hypothesis,
when you should reject it! That’s bad! The
____________________________________ of that is a new term called
POWER! Power -
____________________________________________________________________________
• How can you increase the power of any hypothesis test?
o ______________________________________
o ______________________________________ Practice:
In the past, the mean score of the seniors at South High on the
American College Testing (ACT) college entrance examination has
been 20. This year a special preparation course is offered, and all
53 seniors planning to take the ACT test enroll in the course. The
principal believes that the new course will improve the students’
ACT scores. A hypothesis was performed at the α = .01 level and
found the power to be 0.68. a. State the hypotheses and define the
parameter of interest. b. Describe the Type I and Type II error in
this situation. c. Explain what the power of 0.68 means in this
situation. d. Calculate the probability of Type I Error and Type II
Error. e. How could you increase the power of the test?
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12.1 – Carrying out a Significance Test for p [VIDEO #5]
In addition to performing the same “three C’s” we did for CI’s,
you must do what?
__________________________________________________________________________________________
“C”onditions: 1. The data must come from a random sample from
the population.
a. YOU MUST WRITE BEFORE DOING ANY CALCULATIONS: “We have a SRS
of (size/#)
(people or whatever you sampled) to represent ALL (people or
whatever you sampled).
b. If there is no mention of a random sample, then write “Sample
not known to be random,
proceed with caution!” and continue with the problem.
i. IF data came from a sampling method of high bias (like
volunteer response), then
mention this information and STOP. It’s not worth calculating
& concluding anything. ii. IF data came from a sampling method
not necessarily associated with high bias (like a
convenience sample), then mention this and say “Proceed with
caution!”
2. The observations of your sample must be independent. a. Check
the “10%” condition a.k.a. “Population > 10·n”. SHOW YOUR
WORK!
If it checks, then write “We may calculate the standard
error.”
b. If NOT, then you need to use the Finite Population Correction
formula we mentioned before but will not use in this class, so
write “Standard error cannot be safely calculated.” STOP.
3. The sampling distribution of 𝑝 must be approximately normal.
a. Check the condition n· 𝑝! > 10 and n(1 - 𝑝!) > 10 where 𝑝!
is the hypothesized value for p.
SHOW THE ACTUAL WORK HERE!
If it checks, then write “We may assume the sampling
distribution is approx. normal.”
b. If NOT, then write “Sampling distribution not normal,
hypothesis test cannot be
performed.” STOP.
c. NOTE: The CLT does NOT work with proportion problems, so
don’t use it here!!! “C”alculations: In video #2, we discussed the
concept of a p-value (total evidence against the defendant) in
comparison to the significance level (the tipping point where the
jury goes from thinking the defendant is not guilty to guilty). If
the p-value was smaller (or more extreme than) the significance
level, we rejected Ho in favor of Ha. We now need to discuss how to
calculate that p-value! To get the p-value we need a test
statistic.
We will get into the specifics of this formula in the next
video, but just know this for now…
The “test statistic” is really a _____________ that says how
“far away” our actual sample value is from our hypothesized
population value. If it’s “far enough” away, then Ho is more than
likely wrong!
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When we did CI’s, I told you to not worry about the negative you
would get with invNorm or invT, but you DO need to keep whatever
sign you get when calculating the test statistic! We no longer just
“±” it!!! Now, the z-score is from a
_________________________________________________________________,
which has a mean of ______ and standard deviation of ________. The
p-value is the area under the curve “as extreme as or more extreme
than” this z-score (test statistic). Remember how to find the area
under the curve with lower and upper bounds???
_______________________ Draw a normal curve, find the approximate
spot for the z-score, and then shade…WHICH WAY??? The shading
depends on the ___________________________________________. If it
uses ____, then shade to the ___________ of the z-score. If it uses
_____, then shade to the __________ of the z-score. If it uses
_____, then we will shade to the outside of both the positive and
negative version of the z-score. Thus, we will need to
_________________ our p-value in the end! “C”onclusion: And if the
p-value is ____ our pre-determined α-level, then we get to reject
Ho in favor of Ha!!! Example You want to test the notion of
“home-court advantage”, that the home team wins more than half of
all games. Out of 30 NBA games between January 3rd and 7th 1992, 18
were won by the home team. We will assume all conditions have been
met. The standard deviation of the statistic is 0.09129 (you will
calculate this value directly in the next video). State the null
and alternate hypotheses, calculate the test statistic and p-value.
What would you conclude at the 5% significance level?
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12.1 – One-sided z-Test for p [VIDEO #6] Why is this entitled
“one-sided”??? When the alternate hypothesis uses ____ or ____,
then we are only considering ONE SIDE of the test statistic
(z-score). We will discuss a “two-sided” test in the next
video!!!
“C”onditions: 1. The data must come from a random sample from
the population.
a. YOU MUST WRITE BEFORE DOING ANY CALCULATIONS: “We have a SRS
of (size/#)
(people or whatever you sampled) to represent ALL (people or
whatever you sampled).
b. If there is no mention of a random sample, then write “Sample
not known to be random,
proceed with caution!” and continue with the problem.
i. IF data came from a sampling method of high bias (like
volunteer response), then
mention this information and STOP. It’s not worth calculating
& concluding anything.
ii. IF data came from a sampling method not necessarily
associated with high bias (like a
convenience sample), then mention this and say “Proceed with
caution!”
2. The observations of your sample must be independent.
a. Check the “10%” condition a.k.a. “Population > 10·n”. SHOW
YOUR WORK! If it checks, then write “We may calculate the standard
error.”
b. If NOT, then you need to use the Finite Population Correction
formula we mentioned before but
will not use in this class, so write “Standard error cannot be
safely calculated.” STOP.
3. The sampling distribution of 𝑝 must be approximately
normal.
a. Check the condition n · 𝑝! > 10 and n(1 - 𝑝! ) > 10
where 𝑝! is the hypothesized value for p. SHOW THE ACTUAL WORK
HERE!
If it checks, then write “We may assume the sampling
distribution is approx. normal.”
b. If NOT, then write “Sampling distribution not normal,
hypothesis test cannot be
performed.” STOP.
c. NOTE: The CLT does NOT work with proportion problems, so
don’t use it here!!! “C”alculations: In the last video, we
discussed the general formula to calculate the test statistic:
which will really turn into…
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The p-value is the area under the curve “as extreme as or more
extreme than” this z-score (test statistic). Remember how to find
the area under the curve with lower and upper bounds???
_______________________ “C”onclusion: And if the p-value is ____
our pre-determined α-level, then we get to reject Ho in favor of
Ha!!!
• Here’s what you write as your concluding statement: “We
(circle one à ) do / do not have enough
evidence at the (α) level to conclude that the (population
parameter in context) for all (what/who
is the population in context) is (state the alternate hypothesis
here).”
• The conclusion statement ALWAYS refers to the alternate
hypothesis, ALWAYS!!!
Example In the case of Hazelwood School District v. United
States (1977), the U.S. Government used the City of Hazelwood, a
suburb of St. Louis, on the grounds that it discriminated against
blacks in its hiring of school teachers. The statistical evidence
introduced notes that of the 405 teachers hired in 1972 and 1973
(the years following the passage of the Civil Rights Act), only 15
had been black. The proportion of black teachers living in the
county of St. Louis at the time was 5.7% if one does not include
the city of St. Louis. State the hypotheses & calculate the
test statistic and p-value. What conclusion would you make at the
5% significance level?
Question: Do I have to write down the test statistic (z-score)
to receive full credit on problems when all we really need and use
is the p-value to determine if we reject the null hypothesis??
Answer: YES. This is required on the AP Stats exam, so I will
require you to write it down, too!
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12.1 – Two-sided z-Test for p [VIDEO #7]
2-Sided Test 1-Sided Test (Lower) 1-Sided Test (Upper)
1. State Hypothesis
Null Hypothesis H0: p = ____ H0: p = ____ H0: p = ____
Alternate Hypothesis Ha: p ≠ ____ Ha: p < ____ Ha: p >
____
2. Conditions 1. Do we have a random sample [SRS]? 2. Are the
observations independent [Pop > 10n à 𝝈𝒑 ok to calculate]? 3. Is
the sampling distribution approx. Normal? [np & n(1-p) > 10
check]
3. Calculations
Test Statistic
𝑧 =𝑝 − 𝑝!𝑝!(1− 𝑝!)
𝑛
P-value
4. Conclusion Always refers to Ha in the end!!! Make sure it is
in context to the problem!!!
When you do not know which way the alternate value could go,
then we will perform a two-sided test with ≠.
Our test statistic could end up positive or negative, we just do
not know.
One thing you CANNOT do is look at your data afterwards and see
which direction the test statistic is at
compared to the hypothesized value and switch to a one-sided
test…that’s statistical cheating!!!
One more thing to consider when performing a two-sided test: you
now have two __________ to calculate the
area of to represent the ______________________. Since the
normal distribution is ALWAYS
___________________________, then you simply have to
___________________ one tail’s area!!
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Example A coin that is balanced should come up heads half the
time in the long run. The French naturalist Count Buffon
(1707-1788) wanted to test this theory. He tossed a coin 4040
times. He got 2084 heads. Is this evidence that Buffon’s coin was
not balanced!?!?! Perform a significance test at the 5%
α-level.
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Why CI’s give more information [VIDEO #8] Consider the example
from last video testing whether a coin was fair or not. We
concluded that the coin was not fair because the p-value was less
than the α-level. We know head’s comes up more often than tails as
a result, but it does not give us any idea as to what the true
proportion of flipping heads is for all instances.
Looking at the calculator output for a 95% CI and for a 5%
hypothesis test reveals a special connection:
Notice: Is the hypothesized value of p = 0.5 being captured by
the CI? ________ There will ALWAYS be this special connection
between a CI and a two-sided hypothesis test. There is a connection
between a CI and a one-sided test, but it’s not as clear cut as it
is with a two-sided test. Example A friend of yours recently got a
73% on his last AP Stats test. He thinks the class average is about
the same as his grade. You disagree with his assumption, but
honestly have no idea if the class average is really higher or
lower than his grade. Below are computer print-outs of a 95% CI and
a hypothesis test at the 5% α-level based on a sample of 40
randomly selected test scores from the last test. What conclusion
would you make regarding Ho = 0.73 vs. Ha ≠ 0.73?
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11.1 – Carrying out a Significance Test for µ [VIDEO #9]
In addition to performing the same “three C’s” we did for CI’s,
you must do what?
__________________________________________________________________________________________
“C”onditions: 1. The data must come from a random sample from
the population.
a. YOU MUST WRITE BEFORE DOING ANY CALCULATIONS: “We have a SRS
of (size/#)
(people or whatever you sampled) to represent ALL (people or
whatever you sampled).
b. If there is no mention of a random sample, then write “Sample
not known to be random,
proceed with caution!” and continue with the problem.
i. IF data came from a sampling method of high bias (like
volunteer response), then
mention this information and STOP. It’s not worth calculating
& concluding anything.
ii. IF data came from a sampling method not necessarily
associated with high bias (like a convenience sample), then mention
this and say “Proceed with caution!”
2. The observations of your sample must be independent.
a. Check the “10%” condition a.k.a. “Population > 10·n”. SHOW
YOUR WORK! If it checks, then write “We may calculate the standard
error.”
b. If NOT, then you need to use the Finite Population Correction
formula we mentioned before but will not use in this class, so
write “Standard error cannot be safely calculated.” STOP.
3. The sampling distribution of 𝑥 must be approximately normal.
***If you know σ somehow, then you can say and use the normal
distribution (z-scores).
***If you do not know σ, then you can saw and use the
t-distribution (t-scores). (More likely)
a. If the population is known to be Normal (REGARDLESS OF SAMPLE
SIZE), then write “Since
the population is known to be normal, then we may assume the
sampling distribution can
follow an (select one: approx. normal or t ) distribution.”
b. If NOT, then check for a sufficiently large enough sample
size (> 30) for the CLT to apply
If so, write “Sampling distribution can use t-procedures due to
the CLT and sample size.”
c. If sample size is less than 30: t procedures can be used
except in the presence of outliers or
strong skewness (some skewness OK). Plot your data (stemplot or
boxplot) to check for
Normality (roughly symmetric, single peak, no outliers). If the
data checks out, then write
“Based on the data shown, we may assume a t-distribution with
___ df.”
d. If the data are STRONGLY skewed or if outliers are present,
do not use t. STOP.
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“C”alculations: The “generic” formula that we used for
proportion hypothesis tests: à turns into this for mean hypothesis
tests… The “test statistic” is really a _____________ that says how
“far away” our actual sample value is from our hypothesized
population value. If it’s “far enough” away, then Ho is more than
likely wrong! The p-value is the area under the curve “as extreme
as or more extreme than” this t-score (test statistic). Remember
how to find the area under the curve with lower and upper bounds???
_______________________ Draw a t-curve, find the approximate spot
for the t-score, and then shade…WHICH WAY??? The shading depends on
the ___________________________________________. If it uses ____,
then shade to the ___________ of the t-score. If it uses _____,
then shade to the __________ of the t-score. If it uses ____, then
label both the positive AND negative t-scores and shade to the
____________ of those two t-scores. “C”onclusion: And if the
p-value is ____ our pre-determined α-level, then we get to reject
Ho in favor of Ha!!! Example Your favorite radio station claims to
play an average of 50 minutes of music every hour. However, it
seems that every time you turn to this station, there is a
commercial playing. To investigate their claim, you randomly select
12 different hours during the next week and record what the radio
station plays in each of the 12 hours. The sample mean, 𝑥, from
your data is 47.9 minutes with a sample standard deviation, s, of
2.81 minutes. Assume the conditions have been met.
1. What must be true about our data to conclude it passed the
normality check?
2. Calculate the test statistic.
3. Calculate the p-value.
4. What conclusion can you make at the 5% significance level? 1%
α-level?
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11.1 – One-sided t-Test for µ [VIDEO #10] When performing a
one-sided hypothesis test, keep in mind that the sign [negative or
positive] that we get from calculating our test statistic IS NOW
important. We don’t chop off the negative like we did with CI’s!!!
Step #1: State hypotheses and define the parameter of interest.
Step #2: “C”onditions Step #3: “C”alculations Step #4: “C”onclusion
Example Caffeine, a chemical found in many popular beverages, is
known for reducing fatigue. A college student wants to investigate
his daily caffeine intake from beverages such as soft drinks,
energy drinks, tea, and coffee… beverages this particular student
consumes daily. If he becomes convinced that he consumes over 500
mg of caffeine per day then he will change his habits. Here is his
daily caffeine consumption (in mg) for 14 days:
720 550 340 450 280 700 750 1050 380 1050 400 350 800 900
Perform a significance test at the α = 0.05 level.
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11.2 – Two-sided t-Test for µ [VIDEO #11] Step #1: State
hypotheses and define the parameter of interest. Step #2:
“C”onditions Step #3: “C”alculations Step #4: “C”onclusion The only
BIG difference we will have to do for a two-sided vs. a one-sided
test is to ____________ the p-value! Example In the children’s game
Don’t’ Break the Ice, small plastic ice cubes are squeezed into a
square frame. Each child takes turns tapping out a cube of “ice”
with a plastic hammer, hoping that the remaining cubes don’t
collapse. For the game to work correctly, the cubes must be big
enough so that they hold each other in place in the plastic frame
but not so big that they are too difficult to tap out. The machine
that produces the plastic cubes is designed to make cubes that are
29.5mm wide, but the actual width varies a little. To ensure that
the machine is working well, a supervisor inspects a random sample
of 50 cubes every hour and measures their width. Use the
information from the print-out to test this claim. Also, verify
your conclusion with the 95% CI.
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19
11.2 – Matched Pairs t-Test for µ [VIDEO #12] Police trainees
were seated in a darkened room facing a projector screen. Ten
different license planes were projected on the screen, one at a
time, for 5 seconds each, separated by 15-second intervals.
After the last 15-second interval, the lights were turned on and
the police trainees were asked to write down as many of the 10
license plate numbers as possible, in any order at all.
A random sample of 15 trainees who took this test was then given
a week-long memory training course. They were then retested.
Test, at the 5% significance level, whether the memory course
improved the ability of the trainees to correctly identify license
plates.
Officer Plates correctly Identified before
training
Plates correctly Identified after
training 1 6 6 2 5 8 3 6 6 4 5 7 5 7 9 6 5 8 7 4 9 8 6 6 9 7
7
10 8 5 11 4 9 12 5 8 13 4 6 14 6 8 15 7 6
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20
Upon completion of Chapters 10 - 12, you should be able to: !
Determine whether a specific p-value is significant at the .05 or
.01 level, both or neither level. ! Find a p-value given a
hypothesis and a test statistic. ! Carry out a hypothesis test from
a CI. Remember, the alternative hypothesis must be two sided and
the α
level must be complementary with the CI level. See your notes if
you forget how to state the conclusion when carrying out a test
from a CI.
! Explain the differences between Type I Error, Type II Error,
and Power. Know the meaning of each and that Type I Error is just
α, and that Type II Error and Power are complementary. Also know
that you must have a fixed α in advance and that values far from Ho
are easier to detect (higher power) than those close to Ho.
! Carry out a complete hypothesis test. Remember EVERY component
of a hypothesis test. ! Set up the null and alternative hypothesis
and define µ for a hypothesis test. ! Check conditions. ! Calculate
the test statistic and p-value. Remember – Double the p-value for
2-sided tests. ! Write a concluding statement for a hypothesis test
in the context of the problem. ! Describe Type I and Type II error
in regards to the hypothesis test you carry out. ! Reduce the
probability of Type II Error and increase the power of a test. What
can be done to obtain high
power (low type II error) for any hypothesis test? ! Make sure
you know your formulas and don’t get the components of them mixed
up. You should know
them off by heart from doing HW problems. ! How to verify that
the population is normal with either a stem-plot or a histogram, be
sure to note the
absence of outliers and strong skewness ! How to increase the
power of any test ! Tell the difference between a one sample t-test
and a matched pairs t-test ! Calculate the standard error for both
a one sample proportion. ! Interpret the p-value of a hypothesis
test