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Copyright 2013, 2009, and 2007, Pearson Education, Inc.Chapter 7Sampling DistributionsSection 7.1How Sample Proportions Vary Around the Population Proportion1Example: 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.#2 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 PollsCopyright 2013, 2009, and 2007, Pearson Education, Inc.#3 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 PollsCopyright 2013, 2009, and 2007, Pearson Education, Inc.#4 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 PollsCopyright 2013, 2009, and 2007, Pearson Education, Inc.#5Figure 7.1 The population (9.5 million voters) and data (n=3889) distributions ofcandidate 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.#6Sampling DistributionThe 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 PollsCopyright 2013, 2009, and 2007, Pearson Education, Inc.#7Describing the Sampling Distribution ofa 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.#8Summary 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.#9Copyright 2013, 2009, and 2007, Pearson Education, Inc.Chapter 7Sampling DistributionsSection 7.2How Sample Means Vary Around the Population Mean10How Sample Means Vary Around the Population Mean There are two main results about the sampling distribution of the sample mean:

One result provides formulas for its mean and standard deviation of the sampling distribution.

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.#11Describing 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 , 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.#12The 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.#13Population DistributionFigure 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.#14Example: Weekly Mean SalesAunt Ermas 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.#15The 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.#16Figure 7.9 portrays a possible population distribution for daily sales that is somewhat symmetric and unimodal.Example: Weekly Mean SalesFigure 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.#17Effect of n on the Standard Deviationof the Sampling DistributionWith larger samples, the sample mean tends to fall closer to the population mean.Lets 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.#18Copyright 2013 Pearson Education, Inc. 7.1 Suppose that 40% of men over the age of 30 suffer from lower back pain. For a random sample of 50 men over the age of 30, find the mean and the standard error of the sampling distribution of the sample proportion of men over the age of 30 that suffer from lower back pain. a) Mean = 0.40 Standard Error = 0.0693b) Mean= 20 Standard Error = 3.464c) Mean = 0.40 Standard Error = 3.464d) Mean = 20 Standard Error = 0.0693e) Cannot be determined

Copyright 2013 Pearson Education, Inc. 7.2 Suppose that 40% of men over the age of 30 suffer from lower back pain. For a random sample of 50 men over the age of 30 find the mean and the standard deviation of X (the number of men over the age of 30 that suffer from lower back pain.)

a) Mean = 0.40 Standard Error = 0.0693b) Mean = 20 Standard Error = 3.464c) Mean = 0.40 Standard Error = 3.464d) Mean = 20 Standard Error = 0.0693e) Cannot be determinedCopyright 2013 Pearson Education, Inc. 7.3 Suppose that a pre-election poll of 500 people showed that 51% of the sample supported the incumbent senator. If the population proportion who supported the incumbent senator is really 48%, how likely is it that we would see poll results such as this or higher? a) 0.006b) 0.03c) 0.0901d) 0.9099e) 0.9680Copyright 2013 Pearson Education, Inc. 7.4 Suppose that 80% of Americans prefer milk chocolate to dark chocolate. Is the sampling distribution of the sample proportion that prefers milk chocolate approximately normally distributed for samples of size 200? a) Yes, because n is bigger than 30. b) Yes, because n is bigger than 15. c) Yes, because and .d) No, because or is not greater than 15.

Copyright 2013 Pearson Education, Inc. 7.5 What is the sampling distribution of the sample proportion if and ? a) Approximately Normal with a mean of p and a standard error of

b) Approximately Normal with a mean of np and a standard error of

c) Approximately Binomial with a mean of p and a standard error of

d) Approximately Binomial with a mean of np and a standard error of

Copyright 2013 Pearson Education, Inc. 7.6 Suppose that you and 100 other people ask 25 randomly selected workers how much money they spent on lunch. Which of the following statements would be true?a) All samples would result in the same sample mean.

b) All samples would results in slightly different sample means. Copyright 2013 Pearson Education, Inc. 7.7 Suppose that you wanted to take a sample of South Carolina elementary school teachers. What impact does using a larger sample size have on the sampling distribution of ?a) The mean will increase.b) The mean will decrease.c) The standard error will increase. d) The standard error will decrease.

Copyright 2013 Pearson Education, Inc. 7.8 Suppose that South Carolina elementary school teacher salaries have a distribution that is right skewed with a mean of $27,000 and a standard deviation of $2,000. Suppose that someone took a random sample of 40 elementary school teachers salaries and found the sample mean. What is the standard error of ?

Copyright 2013 Pearson Education, Inc. 7.9 Suppose that for people in Idaho the population mean number of hours worked per week is 40.2 hrs and the population standard deviation is 0.4 hrs. Between what two values will 95% of all sample means from all possible samples of size 40 lie between?a) (38.94, 41.47)b) (39.40, 41.00)c) (40.07, 40.33)d) (40.14, 40.26)

Copyright 2013 Pearson Education, Inc. 7.10 For which combination of population and sample size listed below will you find the sampling distribution of the sample mean approximately normally distributed?a) Population is Right Skewed and n = 10 b) Population is Right Skewed and n = 40 c) Population is Bell Shaped and n = 10 d) B and C onlye) A, B and CCopyright 2013 Pearson Education, Inc. 7.11 True or False: For one population distribution there is only one data distribution.a) True

b) FalseCopyright 2013 Pearson Education, Inc. 7.12 With larger sample sizes there is a greater likelihood that the data distributiona) will look similar to the population distribution.b) will look less like the population distribution.c) is the same as the sampling distribution of the sample mean.d) is the same as the sampling distribution of the sample proportion.Copyright 2013 Pearson Education, Inc. 7.13 The distribution of textbook sales for all college students is right (Rt.) skewed with a mean of $300 and a standard deviation of $120. Suppose that a researcher who didnt know this information sampled 40 students. She found that the students paid $280 on average with a standard deviation equal to $109. What is the population distribution? a) Shape: Normal Mean: 300 Stdev:b) Shape: Approx. Normal Mean: 300 Stdev: c) Shape: Rt. Skewed Mean: 300 Stdev: d) Shape: Rt. Skewed Mean: 280 Stdev:

Copyright 2013 Pearson Education, Inc. 7.14 The distribution of textbook sales for all college students is right (Rt.) skewed with a mean of $300 and a standard deviation of $120. Suppose that a researcher who didnt know this information sampled 40 students. She found that the students paid $280 on average with a standard deviation equal to $109. What is the data distribution?a) Shape: Approx. Normal Mean: 300 Stdev: b) Shape: Most likely Rt. Skewed Mean: 280 Stdev:c) Shape: Most likely Rt. Skewed Mean: 300 Stdev: d) Shape: Approx. Rt. Skewed Mean: 300 Stdev:

Copyright 2013 Pearson Education, Inc. 7.15 The distribution of textbook sales for all college students is right (Rt.) skewed with a mean of $300 and a standard deviation of $120. Suppose that a researcher who didnt know this information sampled 40 students. She found that the students paid $280 on average with a standard deviation equal to $109. What is the sampling distribution of the sample mean for a sample of size 40? a) Shape: Approx. Normal Mean: 300 Stdev:b) Shape: Approx. Normal Mean: 280 Stdev: c) Shape: Approx. Normal Mean: 300 Stdev:d) Shape: Approx. Normal Mean: 300 Stdev: