1)Find a distribution of 5 positive whole numbers with the following properties: a)Mean = 4 b)Median = 3 c)Mode = 3 2)How many distributions can you find.

1)Find a distribution of 5 positive whole numbers with the following properties:a) Mean = 4b) Median = 3c) Mode = 3

2)How many distributions can you find with these same properties?

3)What is the maximum range of the distribution?4)Find the distribution of 5 positive whole numbers

with a mean = 31, median = 33, Mode = 34, Range = 8

Math I

UNIT QUESTION: How do you use probability to make plans and predict for the future?Standard: MM1D1-3

Today’s Question:How do we take a random sample, and what statistics can we find with the data?Standard: MM1D3.a.

Data consists of information coming from observations, counts, measurement, or responses. The singular of data is datum.

Statistics is the science of collecting, organizing, analyzing, and interpreting data in order to make decisions.

A population is the collection of all outcomes, responses, measurements, or count that are of interest

A sample is a subset of a population

A parameter is a numerical description of a population characteristic.

A statistic is numerical description of a sample characteristic

Population Parameter Sample Statistic

Give some examples of why we would want to use data from a sample instead of looking at the whole population

Sampling can be less expensive

Sampling can be used to make predictions of things that have not happened yet – think of polls in an election

Sometimes the sample is destroyed. Think of bombs or destructive testing of engineering materials. If we were to sample the whole population, it would all be destroyed.

Reason: Easy and less expensive

Problems with non-statisical sampling is that the results can be misleading.

A voluntary sample is made up of people who self-select into the survey. Often, these folks have a strong interest in the main topic of the survey. Give some examples.

A convenience sample is made up of people who are easy to reach. Give some examples.

Simple Random Sampling (SRS): All data points have an equal chance of being selected. We will be using this one the most.

Other common types of sampling include:

1.Census – count the whole population

2.Stratified sampling – cut the population into parts (strata) and then select sample randomly from each part

3.Cluster sampling – cut the population into parts (clusters), then sample all points within the selected clusters

4.Systematic random sampling – sample every nth data point

A biased sample is a sample that is not representative of the population.

A representative sample is a sample that accurately reflects the characteristics of the population.

Some Causes of Bias• Undercoverage. Undercoverage occurs when some

members of the population are inadequately represented in the sample.

• A classic example of undercoverage is the Literary Digest voter survey, which predicted that Alfred Landon would beat Franklin Roosevelt in the 1936 presidential election.

• The survey relied on a convenience sample, drawn from telephone directories and car registration lists. In 1936, people who owned cars and telephones tended to be more affluent. Undercoverage is often a problem with convenience samples.

• The survey sample suffered from undercoverage of low-income voters, who tended to be Democrats.

Some Causes of Bias

• Nonresponse bias. Sometimes, individuals chosen for the sample are unwilling or unable to participate in the survey.

Some Causes of Bias

• Voluntary response bias occurs when sample members are self-selected volunteers.

• An example would be call-in radio shows that solicit audience participation in surveys on controversial topics (abortion, affirmative action, gun control, etc.). The resulting sample tends to overrepresent individuals who have strong opinions.

Some Causes of Bias

• Leading questions. The wording of the question may be loaded in some way to unduly favor one response over another.

• Example: Would you favor raising taxes to help feed our starving children?

• Example: You like my outfit, don’t you?

Some Causes of Bias• Social desirability. Most people like to

present themselves in a favorable light, so they will be reluctant to admit to unsavory attitudes or illegal activities in a survey, particularly if survey results are not confidential.

• Example: How many times have you done drugs in the past 30 days?

• Example: How many times have you kicked your baby brother this week?

John surveys every fifth person leaving a pet supply store. Of those surveyed, ¾ support the city manager’s proposition to tear down the old library and replace the area with the construction of a new pet park. John plans to write a letter to the editor of the local newspaper about the proposal for the new pet park stating that there is tremendous support from the citizens of the town for constructing a new pet park.

a) Can the conclusion John stated be accurately supported?

b) Suggest another plan for obtaining a good sample population.

We want to know the average height of students at Salem High School.

What would be a good method to find this number without checking every student?

Sample Problem #1:

Your school is conducting a survey on a proposed dress code. The first 20 students entering the school in the morning are surveyed. What type of sampling method is used?

a)Convenienceb)Randomc)Stratified randomd)Systematic

What could be done to help reduce (notice I did not say eliminate) bias and make the survey a better random sample?

Sample Problem #2:

A principal is surveying the parents of his school to find out if they support extending the school day by 20 minutes. Which of the following methods of choosing a sample will most likely result in a sample that is representative of all parents?

a) Surveying all the parents at a parent – teacher meetingb) Surveying 50 parents of 9th grade studentsc) Surveying all the parents who volunteer for a school fundraiserd) Selecting every 20th name from an alphabetical list of all parents and surveying those parents.

Sample Problem #3:

Explain why the following questions are biased or not:

a) Do you prefer creamy macaroni and cheese or bland rice?

b) Don’t you feel the city is wasting money by building that new stadium?

c) Do you prefer shopping online or in the stores?

d)Don’t you agree that the driving age should be raised to 18 to decrease the number of accidents?

Summary Statistics

• We go to all the bother to generate un-biased sampling to generate representative samples.

• Representative samples provide summary statistics that can be used to predict what will happen in the population.

Use Sample Statistics to Predict Population Behavior

Sample Problem 1:

What is the probability of having three children, the first two being male, and the third female?

0.5 * 0.5 * 0.5 = 0.125

Use Sample Statistics to Predict Population Behavior

Sample Problem 2:

The Chicken Clucker Egg Farm has done a random sample and has determined that on the average, one egg per dozen is cracked. What is the probability that any egg in the farm will be cracked?

1/12 = 0.08

Use Sample Statistics to Predict Population Behavior

Sample Problem 3:

Kareem Abdul-Jabbar had a 0.739 free throw average in the 1988/89 season. Based on these statistics, how many free throws would he make in 7 attempts?

7 * 0.739 = 5.2, or 5 free throws

Work

• Page 361, # 1 – 11all• Finish the warm-up for extra credit

Other Work• Quiz Tuesday Test Friday• Match bar chart to box and whisker plot• Algebra• Worksheets from Frameworks• 5 number summary worksheet• Presidential data – box plots – box and

whisker plots