AP Statistics Chapter 18AP Statistics Chapter 18

““Sampling Distributions for Sampling Distributions for Sample Proportions and Sample Sample Proportions and Sample

Means”Means”

What are Sampling Distributions?What are Sampling Distributions?

From a population, we draw a sample.From a population, we draw a sample. From the sample, we collect statistics.From the sample, we collect statistics. From the statistics, we infer information about the From the statistics, we infer information about the

population.population.

To understand the relationship between samples To understand the relationship between samples and the population, we look at the sampling and the population, we look at the sampling distributions – the distribution of values of a distributions – the distribution of values of a statistic taken from all possible samples of a given statistic taken from all possible samples of a given sizesize

The Distribution of Sample The Distribution of Sample ProportionsProportions

If a simple random sample (SRS) of size n is If a simple random sample (SRS) of size n is drawn from a large population with a drawn from a large population with a proportion p, the sampling distribution of the proportion p, the sampling distribution of the sample proportion is approximately normal sample proportion is approximately normal with a mean of p and a standard deviation of with a mean of p and a standard deviation of

n

pq

Sample Proportions

Mean = p

Standard Deviation =

n

pq

Example 1Example 1 Based on past experience, a bank believes that 7% of the Based on past experience, a bank believes that 7% of the

people who receive loans will not make payments on time. The people who receive loans will not make payments on time. The bank has recently approved 200 loans. What are the mean and bank has recently approved 200 loans. What are the mean and standard deviation of the proportion of clients in this group who standard deviation of the proportion of clients in this group who may not make timely payments? Draw the normal model. Label may not make timely payments? Draw the normal model. Label up to three standard deviations. What is the probability that over up to three standard deviations. What is the probability that over 10% of these clients will not make timely payments?10% of these clients will not make timely payments?

7%

Mean = 7%

STD = %8.1

200

%93%7

n

pq

8.8%5.2%10.6%3.4%

12.4%1.6%P(over 10%): z-score = 667.1

%8.1

%7%10

STD

meanx

normalcdf (1.667, 99) = 4.8%

Example 2Example 2 Public health statistics indicate that 26.4% of American adults Public health statistics indicate that 26.4% of American adults

smoke cigarettes. Using the normal model, draw the sampling smoke cigarettes. Using the normal model, draw the sampling distribution model for the proportion of smokers among a distribution model for the proportion of smokers among a randomly selected group of 50 adults. (Label your picture up to randomly selected group of 50 adults. (Label your picture up to three standard deviations.) What is the probability that over 33% three standard deviations.) What is the probability that over 33% of this group will smoke?of this group will smoke?

Mean = 26.4%

STD = %2.6

50

%6.73%4.26

n

pq26.4%

32.6%20.2%38.8%14%

45%7.8%P(over 33%): z-score = 0645.1

%2.6

%4.26%33

STD

meanx

normalcdf (1.0645, 99) = 14.4%

Conditions and AssumptionsConditions and Assumptions

If these conditions are not met for your sample, If these conditions are not met for your sample, then the normal model should not be used to then the normal model should not be used to approximate the sampling distribution of the approximate the sampling distribution of the sample proportion:sample proportion:

Must be an independent random sampleMust be an independent random sample The population must be at least 10 times the The population must be at least 10 times the

sample sizesample size np > 10 and nq > 10 (a.k.a. minimum of 10 np > 10 and nq > 10 (a.k.a. minimum of 10

successes and 10 failures in the sample)successes and 10 failures in the sample)

The Distribution of Sample MeansThe Distribution of Sample Means

If a simple random sample (SRS) of size n is If a simple random sample (SRS) of size n is drawn from a large population with a mean drawn from a large population with a mean μμ, , the sampling distribution of the sample mean the sampling distribution of the sample mean is approximately normal with a mean of is approximately normal with a mean of μμ and and a standard deviation of a standard deviation of

n

Sample Means

Mean = μ

Standard Deviation =

n

Example 3Example 3 Assume that the distribution of human pregnancies can be Assume that the distribution of human pregnancies can be

described by a Normal model with mean 266 days and standard described by a Normal model with mean 266 days and standard deviation 16 days. Draw a normal model for the distribution of deviation 16 days. Draw a normal model for the distribution of the mean length of the pregnancies among a sample of 60 the mean length of the pregnancies among a sample of 60 pregnant women at a certain obstetrician’s office. Label your pregnant women at a certain obstetrician’s office. Label your picture up to three standard deviations. What is the probability picture up to three standard deviations. What is the probability that the mean duration of these pregnancies will be less than that the mean duration of these pregnancies will be less than 260 days?260 days?

Mean = 266 days

STD = 07.260

16

n

std266

268.07263.93270.14261.86

272.21259.79P(less than 260 days): z-score = 899.2

07.2

266260

STD

meanx

normalcdf (-99, -2.899) = 0.19%

Conditions and AssumptionsConditions and Assumptions

If these conditions are not met for your sample, If these conditions are not met for your sample, then the normal model should not be used to then the normal model should not be used to approximate the sampling distribution of the approximate the sampling distribution of the sample mean:sample mean:

Must be an independent random sampleMust be an independent random sample The population must be at least 10 times the The population must be at least 10 times the

sample sizesample size The sample size n should be large (more than The sample size n should be large (more than

forty)forty)