Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 7 Sampling Distributions Section 7.1 How Sample Proportions Vary Around the Population Proportion

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 3 Example: Predicting Election Results Using Exit Polls How do we know if the sample proportion from the California exit poll is a good estimate, falling close to the population proportion? The total number of voters was over nine million, and the poll sampled a minuscule portion of them. This section introduces a type of probability distribution called the Sampling Distribution that helps us determine how close to the population parameter a sample statistic is likely to fall.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 4 Using exit polls, polling organizations predict winners after learning how a small number of people voted, often only a few thousand out of possibly millions of voters. After sampling 3889 randomly selected voters, 53.1% said they voted for Brown, 42.4% for Whitman. At the time of the exit poll, the percentage of the entire voting population (nearly 9.5 million people) that voted for Brown was unknown. Example: Predicting Election Results Using Exit Polls

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 5 How close can we expect a sample percentage to be to the population percentage? How does the sample size influence our analysis? The sampling distribution helps us determine how close to the population parameter a sample statistic is likely to fall. Example: Predicting Election Results Using Exit Polls

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 6 Let X = vote outcome, with x = 1 for Jerry Brown and x = 0 for all other responses. The possible values of the random variable X (0 and 1) and how often these values occurred (0.469 and 0.531) give the data distribution for this one sample. The possible values of the random variable X (0 and 1) and how often these values occurred (0.462 and 0.538) give the population distribution. Example: Predicting Election Results Using Exit Polls

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 7 Figure 7.1 The population (9.5 million voters) and data (n=3889) distributions of candidate preference (0 = Not Brown, 1= Brown). Question: Why do these look so similar? Example: Predicting Election Results Using Exit Polls

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 8 Sampling Distribution The sampling distribution of a statistic is the probability distribution that specifies probabilities for the possible values the statistic can take. A sampling distribution is merely a type of probability distribution. Rather than giving probabilities for an observation for an individual subject (as in a population or data distribution), it gives probabilities for the value of a statistic for a sample of subjects. Example: Predicting Election Results Using Exit Polls

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 9 Describing the Sampling Distribution of a Sample Proportion We typically use the mean to describe center and the standard deviation to describe variability. For the sampling distribution of a sample proportion, the mean and standard deviation depend on the sample size n and the population proportion p. For a random sample of size n from a population with proportion p of outcomes in a particular category, the sampling distribution of the sample proportion in that category has

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 10 Summary of Sampling Distribution of a Sample Proportion If n is sufficiently large so that the expected numbers of outcomes of the two types, np in the category of interest and n(1 - p) not in that category, are both at least 15, then the sampling distribution of a sample proportion is approximately normal.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 7 Sampling Distributions Section 7.2 How Sample Means Vary Around the Population Mean

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 12 How Sample Means Vary Around the Population Mean There are two main results about the sampling distribution of the sample mean: 1.One result provides formulas for its mean and standard deviation of the sampling distribution. 2.The other indicates that its shape is often approximately a normal distribution, as we observed in the previous section for the sample proportion.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 13 Describing the Behavior of the Sampling Distribution for the Sample Mean for any Population Even when a population distribution is not bell shaped, the sampling distribution of the sample mean can have a bell shape. We also observe that the mean of the sampling distribution of the sample mean appears to be the same as the population mean μ, and the standard deviation of the sampling distribution for the sample mean appears to be: This bell shape is a consequence of the central limit theorem (CLT).

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 14 The Central Limit Theorem (CLT): Describes the Expected Shape of the Sampling Distribution for Sample Mean For a random sample of size n from a population having mean and standard deviation, then as the sample size n increases, the sampling distribution of the sample mean approaches an approximately normal distribution.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 15 Population Distribution Figure 7.8 Four Population Distributions and the Corresponding Sampling Distributions of. Regardless of the shape of the population distribution, the sampling distribution becomes more bell shaped as the random sample size n increases.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 16 Example: Weekly Mean Sales Aunt Erma’s Restaurant in the North End of Boston specializes in pizza that is baked in a wood-burning oven. The sales of food and drink in this restaurant vary from day to day. Past records indicate that the daily sales follow a probability (population) distribution with a mean of and a standard deviation of. 1. What would we expect the weekly sample mean sales amounts to fluctuate around (in dollars)? 2. How much variability would you expect in the weekly sample mean sales figures? Find the standard deviation of the sampling distribution of the sample mean, and interpret this standard deviation.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 17 The mean of the assumed population distribution,. The sampling distribution of the sample mean for n = 7 has mean $900. Its standard deviation equals Example: Weekly Mean Sales

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 18 Figure 7.9 portrays a possible population distribution for daily sales that is somewhat symmetric and unimodal. Example: Weekly Mean Sales Figure 7.9 A Population Distribution for Daily Sales and the Sampling Distribution of Weekly Mean Sales. There is more variability day to day in the daily sales than week to week in the weekly mean sales.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 19 Effect of n on the Standard Deviation of the Sampling Distribution With larger samples, the sample mean tends to fall closer to the population mean. Let’s consider again the formula for the standard deviation of the sample mean: Notice that as the sample size n increases, the denominator increases, so the standard deviation of the sample mean decreases. Again, with larger samples, the sample mean tends to fall closer to the population mean.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 7 Sampling Distributions Section 7.3 The Binomial Distribution Is a Sampling Distribution (Optional)

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 21 Mean and Standard Deviation of Sampling Distribution of a Proportion For a binomial random variable with n trials and probability p of success for each trial, the sampling distribution of the proportion of successes has To obtain these values using the binomial distribution, take the binomial mean np and binomial standard deviation of the number of successes and divide by n.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 22 Figure 7.11 The Sampling Distribution of the Sample Proportion Is Less Spread Out Than the Sampling Distribution of the Number of Successes. Mean and Standard Deviation of Sampling Distribution of a Proportion