“Sampling Distributions for Sample Proportions and Sample Means” AP Statistics Chapter 18 “Sampling Distributions for Sample Proportions and Sample Means”

What are Sampling Distributions? From a population, we draw a sample. From the sample, we collect statistics. From the statistics, we infer information about the population. To understand the relationship between samples and the population, we look at the sampling distributions – the distribution of values of a statistic taken from all possible samples of a given size

The Distribution of Sample Proportions If a simple random sample (SRS) of size n is drawn from a large population with a proportion p, the sampling distribution of the sample proportion is approximately normal with a mean of p and a standard deviation of Sample Proportions Mean = p Standard Deviation =

Example 1 Based on past experience, a bank believes that 7% of the people who receive loans will not make payments on time. The bank has recently approved 200 loans. What are the mean and standard deviation of the proportion of clients in this group who may not make timely payments? Draw the normal model. Label up to three standard deviations. What is the probability that over 10% of these clients will not make timely payments? Mean = 7% STD = 7% 5.2% 8.8% 3.4% 10.6% P(over 10%): z-score = 1.6% 12.4% normalcdf (1.667, 99) = 4.8%

Example 2 Public health statistics indicate that 26.4% of American adults smoke cigarettes. Using the normal model, draw the sampling distribution model for the proportion of smokers among a randomly selected group of 50 adults. (Label your picture up to three standard deviations.) What is the probability that over 33% of this group will smoke? Mean = 26.4% STD = 26.4% 20.2% 32.6% 14% 38.8% P(over 33%): z-score = 7.8% 45% normalcdf (1.0645, 99) = 14.4%

Conditions and Assumptions If these conditions are not met for your sample, then the normal model should not be used to approximate the sampling distribution of the sample proportion: Must be an independent random sample The population must be at least 10 times the sample size np > 10 and nq > 10 (a.k.a. minimum of 10 successes and 10 failures in the sample)

The Distribution of Sample Means If a simple random sample (SRS) of size n is drawn from a large population with a mean μ, the sampling distribution of the sample mean is approximately normal with a mean of μ and a standard deviation of Sample Means Mean = μ Standard Deviation =

Example 3 Assume that the distribution of human pregnancies can be described by a Normal model with mean 266 days and standard deviation 16 days. Draw a normal model for the distribution of the mean length of the pregnancies among a sample of 60 pregnant women at a certain obstetrician’s office. Label your picture up to three standard deviations. What is the probability that the mean duration of these pregnancies will be less than 260 days? Mean = 266 days STD = 266 263.93 268.07 261.86 270.14 P(less than 260 days): z-score = 259.79 272.21 normalcdf (-99, -2.899) = 0.19%

Conditions and Assumptions If these conditions are not met for your sample, then the normal model should not be used to approximate the sampling distribution of the sample mean: Must be an independent random sample The population must be at least 10 times the sample size The sample size n should be large (more than forty)