1. Copyright © 2009 Pearson Education, Inc. 8.1 Sampling Distributions LEARNING GOAL Understand the fundamental ideas of sampling distributions and how the distribution of sample means and the distribution of sample proportions are formed. Also learn the notation used to represent sample means and proportions.

2. Slide 8.1- 2 Copyright © 2009 Pearson Education, Inc. Sample Means: The Basic Idea Table 8.1 lists the weights of the five starting players (labeled A through E for convenience) on a professional basketball team. We regard these five players as the entire population (with a mean of 242.4 pounds). Samples drawn from this population of five players can range in size from n = 1 (one player out of the five) to n = 5 (all five players). With a sample size of n = 1, there are 5 different samples that could be selected: Each player is a sample. The mean of each sample of size n = 1 is simply the weight of the player in the sample.

3. Slide 8.1- 3 Copyright © 2009 Pearson Education, Inc. Figure 8.1 shows a histogram of the means of the 5 samples; it is called a distribution of sample means, because it shows the means of all 5 samples of size n = 1. The distribution of sample means created by this process is an example of a sampling distribution. This term simply refers to a distribution of a sample statistic, such as a mean, taken from all possible samples of a particular size. Figure 8.1 Sampling distribution for sample size n 1.

4. Slide 8.1- 4 Copyright © 2009 Pearson Education, Inc. This demonstrates a general rule: The mean of a distribution of sample means is the population mean. Notice that the mean of the 5 sample means is the mean of the entire population: = 242.4 pounds 215 + 242 + 225 + 215 + 315 5

5. Slide 8.1- 5 Copyright © 2009 Pearson Education, Inc. Let’s move on to samples of size n = 2, in which each sample consists of two different players. With five players, there are 10 different samples of size n = 2. Each sample has its own mean. Table 8.2 lists the 10 samples with their means. Figure 8.2 shows the distribution of all 10 sample means. means is equal to the population mean, 242.4 pounds. Again, notice that the mean of the distribution of sample

6. Slide 8.1- 6 Copyright © 2009 Pearson Education, Inc. Ten different samples of size n = 3 are possible in a population of five players. Table 8.3 shows these samples and their means, and Figure 8.3 shows the distribution of these sample means. Again, the mean of the distribution of sample means is equal to the population mean, 242.4 pounds.

7. Slide 8.1- 7 Copyright © 2009 Pearson Education, Inc. With a sample size of n = 4, only 5 different samples are possible. Table 8.4 shows these samples and their means, and Figure 8.4 shows the distribution of these sample means.

8. Slide 8.1- 8 Copyright © 2009 Pearson Education, Inc. Finally, for a population of five players, there is only 1 possible sample of size n = 5: the entire population. In this case, the distribution of sample means is just a single bar (Figure 8.5). Again the mean of the distribution of sample means is the population mean, 242.4 pounds. Figure 8.5 Sampling distribution for sample size n = 5. To summarize, when we work with all possible samples of a population of a given size, the mean of the distribution of sample means is always the population mean.

9. Slide 8.1- 9 Copyright © 2009 Pearson Education, Inc. Sample Means with Larger Populations In typical statistical applications, populations are huge and it is impractical or expensive to survey every individual in the population; consequently, we rarely know the true population mean, μ. Therefore, it makes sense to consider using the mean of a sample to estimate the mean of the entire population. Although a sample is easier to work with, it cannot possibly represent the entire population exactly. Therefore, we should not expect an estimate of the population mean obtained from a sample to be perfect. The error that we introduce by working with a sample is called the sampling error.

10. Slide 8.1- 10 Copyright © 2009 Pearson Education, Inc. Sampling Error The sampling error is the error introduced because a random sample is used to estimate a population parameter. It does not include other sources of error, such as those due to biased sampling, bad survey questions, or recording mistakes.

11. Slide 8.1- 11 Copyright © 2009 Pearson Education, Inc. Results from a survey of students who were asked how many hours they spend per week using a search engine on the Internet. n = 400 μ = 3.88σ = 2.40

12. Slide 8.1- 12 Copyright © 2009 Pearson Education, Inc. A sample of 32 students selected from the 400 on the previous slide. The mean of this sample is x = 4.17; we use the standard notation x to denote this mean. We say that x is a sample statistic because it comes from a sample of the entire population. Thus, x is called a sample mean. 1.1 7.8 6.8 4.9 3.0 6.5 5.2 2.2 5.1 3.4 4.7 7.0 3.8 5.7 6.5 2.7 2.6 1.4 7.1 5.5 3.1 5.0 6.8 6.5 1.7 2.1 1.2 0.3 0.9 2.4 2.5 7.8 Sample 1 ¯ x ¯ x ¯ x ¯ x

13. Slide 8.1- 13 Copyright © 2009 Pearson Education, Inc. Notation for Population and Sample Means n = sample size  = population mean x = sample mean ¯

14. Slide 8.1- 14 Copyright © 2009 Pearson Education, Inc. A different sample of 32 students selected from the 400. Now you have two sample means that don’t agree with each other, and neither one agrees with the true population mean. ¯ x 1.8 0.4 4.0 2.4 0.8 6.2 0.8 6.6 5.7 7.9 2.5 3.6 5.2 5.7 6.5 1.2 5.4 5.7 7.2 5.1 3.2 3.1 5.0 3.1 0.5 3.9 3.1 5.8 2.9 7.2 0.9 4.0 Sample 2 x 1 = 4.17 (slide 13)x 2 = 3.98  = 3.88 (slide 10) ¯ x ¯ x For this sample x is = 3.98.

15. Slide 8.1- 15 Copyright © 2009 Pearson Education, Inc. Figure 8.6 shows a histogram that results from 100 different samples, each with 32 students. Notice that this histogram is very close to a normal distribution and its mean is very close to the population mean, μ = 3.88. Figure 8.6 A distribution of 100 sample means, with a sample size of n = 32, appears close to a normal distribution with a mean of 3.88.

16. Slide 8.1- 16 Copyright © 2009 Pearson Education, Inc. The Distribution of Sample Means The distribution of sample means is the distribution that results when we find the means of all possible samples of a given size. The larger the sample size, the more closely this distribution approximates a normal distribution. In all cases, the mean of the distribution of sample means equals the population mean. If only one sample is available, its sample mean, x, is the best estimate for the population mean, . ¯ x

17. Slide 8.1- 17 Copyright © 2009 Pearson Education, Inc.

18. Slide 8.1- 18 Copyright © 2009 Pearson Education, Inc. The distribution of sample means is approximately a normal distribution. The mean of the distribution of sample means is 3.88 (the mean of the population). The standard deviation of the distribution of sample means depends on the population standard deviation and the sample size. The population standard deviation is σ = 2.40 and the sample size is n = 32, so the standard deviation of sample means is = = 0.42 σnσn 2.40 32 If we were to include all possible samples of size n = 32, this distribution would have these characteristics:

19. Slide 8.1- 19 Copyright © 2009 Pearson Education, Inc. Suppose we select the following random sample of 32 responses from the 400 responses given earlier: The mean of this sample is x = 5.01. 5.8 7.5 5.8 5.2 3.9 3.4 7.3 4.1 0.5 7.9 7.7 7.7 5.0 2.3 7.8 2.3 5.0 6.8 6.5 1.7 2.1 7.3 4.0 2.2 5.6 4.7 5.3 3.5 6.5 3.4 6.6 5.0 Sample 3 ¯ x Given that the mean of the distribution of sample means is 3.88 and the standard deviation is 0.42, the sample mean of x = 5.01 has a standard score of z = = = 2.7 sample mean – pop. mean standard deviation 5.01 – 3.88 0.42 ¯ x

20. Slide 8.1- 20 Copyright © 2009 Pearson Education, Inc. The sample (from the previous slide) has a standard score of z = 2.7, indicating that it is 2.7 standard deviations above the mean of the sampling distribution. From Table 5.1, this standard score corresponds to the 99.65th percentile, so the probability of selecting another sample with a mean less than 5.01 is about 0.9965. It follows that the probability of selecting another sample with a mean greater than 5.01 is about 1 – 0.9965 = 0.0035. Apparently, the sample we selected is rather extreme within this distribution.

21. Slide 8.1- 21 Copyright © 2009 Pearson Education, Inc. Texas has roughly 225,000 farms, more than any other state in the United States. The actual mean farm size is μ = 582 acres and the standard deviation is σ = 150 acres. For random samples of n = 100 farms, find the mean and standard deviation of the distribution of sample means. What is the probability of selecting a random sample of 100 farms with a mean greater than 600 acres? EXAMPLE 1 Sampling Farms Solution

22. Slide 8.1- 22 Copyright © 2009 Pearson Education, Inc. Solution: (cont.) EXAMPLE 1 Sampling Farms

23. Slide 8.1- 23 Copyright © 2009 Pearson Education, Inc. Sample Proportions In a survey where 400 students were asked if they own a car, 240 replied that they did. The exact proportion of car owners is p = = 0.6 240 400 This population proportion, p = 0.6, is another example of a population parameter.

24. Slide 8.1- 24 Copyright © 2009 Pearson Education, Inc. A sample of 32 was selected from the 400 students and 21 were car owners. p = = 0.656 21 32 p ˆ This proportion is another example of a sample statistic. In this case, it is a sample proportion because it is the proportion of car owners within a sample; we use the symbol p (read “p-hat”) to distinguish this sample proportion from the population proportion, p. ˆ p

25. Slide 8.1- 25 Copyright © 2009 Pearson Education, Inc. Notation for Population and Sample Proportions n = sample size p = population proportion p = sample proportion ˆ

26. Slide 8.1- 26 Copyright © 2009 Pearson Education, Inc. Figure 8.7 shows such a histogram of sample proportions from 100 samples of size n = 32. As we found for sample means, this distribution of sample proportions is very close to a normal distribution. Furthermore, the mean of this distribution is very close to the population proportion of 0.6. Figure 8.7 The distribution of 100 sample proportions, with a sample size of 32, appears to be close to a normal distribution.

27. Slide 8.1- 27 Copyright © 2009 Pearson Education, Inc. Suppose it were possible to select all possible samples of size n = 32. The resulting distribution would be called a distribution of sample proportions. The mean of this distribution equals the population proportion exactly. This distribution approaches a normal distribution as the sample size increases. In practice, we often have only one sample to work with. In that case, the best estimate for the population proportion, p, is the sample proportion, p. ˆ p

28. Slide 8.1- 28 Copyright © 2009 Pearson Education, Inc. The Distribution of Sample Proportions The distribution of sample proportions is the distribution that results when we find the proportions ( ) in all possible samples of a given size. The larger the sample size, the more closely this distribution approximates a normal distribution. In all cases, the mean of the distribution of sample proportions equals the population proportion. If only one sample is available, its sample proportion,, is the best estimate for the population proportion, p. ˆ p ˆ p

29. Slide 8.1- 29 Copyright © 2009 Pearson Education, Inc. Consider the distribution of sample proportions shown in Figure 8.7 (slide 30). Assume that its mean is p = 0.6 and its standard deviation is 0.1. Suppose you randomly select the following sample of 32 responses: Y Y N Y Y Y Y N Y Y Y Y Y Y N Y Y N Y Y Y N Y Y N Y Y N Y N Y Y Solution: Compute the sample proportion, p, for this sample. How far does it lie from the mean of the distribution? What is the probability of selecting another sample with a proportion greater than the one you selected? EXAMPLE 2 Analyzing a Sample Proportion ˆ p

30. Slide 8.1- 30 Copyright © 2009 Pearson Education, Inc. Solution: (cont.) EXAMPLE 2 Analyzing a Sample Proportion

31. Copyright © 2009 Pearson Education, Inc. 8.2 Estimating Population Means LEARNING GOAL Learn to estimate population means and compute the associated margins of error and confidence intervals.

32. Slide 8.2- 32 Copyright © 2009 Pearson Education, Inc. Estimating a Population Mean: The Basics When we have only a single sample, the sample mean is the best estimate of the population mean, μ. However, we do not expect the sample mean to be equal to the population mean, because there is likely to be some sampling error. Therefore, in order to make an inference about the population mean, we need some way to describe how well we expect it to be represented by the sample mean. The most common method for doing this is by way of confidence intervals.

33. Slide 8.2- 33 Copyright © 2009 Pearson Education, Inc. A precise calculation shows that if the distribution of sample means is normal with a mean of μ, then 95% of all sample means lie within 1.96 standard deviations of the population mean; for our purposes in this book, we will approximate this as 2 standard deviations. A confidence interval is a range of values likely to contain the true value of the population mean.

34. Slide 8.2- 34 Copyright © 2009 Pearson Education, Inc. 95% Confidence Interval for a Population Mean The margin of error for the 95% confidence interval is where s is the standard deviation of the sample. We find the 95% confidence interval by adding and subtracting the margin of error from the sample mean. That is, the 95% confidence interval ranges from (x – margin of error) to (x + margin of error) We can write this confidence interval more formally as x – E < μ < x + E or more briefly as x ± E margin of error = E ≈ 2s n2s n ¯ x ¯ x ¯ x ¯ x ¯ x

35. Slide 8.2- 35 Copyright © 2009 Pearson Education, Inc.

36. Slide 8.2- 36 Copyright © 2009 Pearson Education, Inc. Compute the margin of error and find the 95% confidence interval for the protein intake sample of n = 267 men, which has a sample mean of x = 77.0 grams and a sample standard deviation of s = 58.6 grams. Solution: EXAMPLE 1 Computing the Margin of Error ¯ x

37. Slide 8.2- 37 Copyright © 2009 Pearson Education, Inc. Solution: (Cont.) EXAMPLE 1 Computing the Margin of Error

38. Slide 8.2- 38 Copyright © 2009 Pearson Education, Inc. Interpreting the Confidence Interval Figure 8.10 This figure illustrates the idea behind confidence intervals. The central vertical line represents the true population mean, μ. Each of the 20 horizontal lines represents the 95% confidence interval for a particular sample, with the sample mean marked by the dot in the center of the confidence interval. With a 95% confidence interval, we expect that 95% of all samples will give a confidence interval that contains the population mean, as is the case in this figure, for 19 of the 20 confidence intervals do indeed contain the population mean. We expect that the population mean will not be within the confidence interval in 5% of the cases; here, 1 of the 20 confidence intervals (the sixth from the top) does not contain the population mean.

39. Slide 8.2- 39 Copyright © 2009 Pearson Education, Inc. Solution: EXAMPLE 2 Constructing a Confidence Interval A study finds that the average time spent by eighth-graders watching television is 6.7 hours per week, with a margin of error of 0.4 hour (for 95% confidence). Construct and interpret the 95% confidence interval. ¯ x

40. Slide 8.2- 40 Copyright © 2009 Pearson Education, Inc. Choosing Sample Size Choosing the Correct Sample Size In order to estimate the population mean with a specified margin of error of at most E, the size of the sample should be at least where σ is the population standard deviation (often estimated by the sample standard deviation s). E E Solve the margin of error formula for n. E ≈ 2s / n

41. Slide 8.2- 41 Copyright © 2009 Pearson Education, Inc. Solution: a. EXAMPLE 6 Constructing a Confidence Interval You want to study housing costs in the country by sampling recent house sales in various (representative) regions. Your goal is to provide a 95% confidence interval estimate of the housing cost. Previous studies suggest that the population standard deviation is about $7,200. What sample size (at a minimum) should be used to ensure that the sample mean is within a. $500 of the true population mean? b. $100 of the true population mean?

42. Slide 8.2- 42 Copyright © 2009 Pearson Education, Inc. Solution: EXAMPLE 6 Constructing a Confidence Interval

43. Copyright © 2009 Pearson Education, Inc. 8.3 Estimating Population Proportions LEARNING GOAL Learn to estimate population proportions and compute the associated margins of error and confidence intervals.

44. Slide 8.3- 44 Copyright © 2009 Pearson Education, Inc. The Basics of Estimating a Population Proportion

45. Slide 8.3- 45 Copyright © 2009 Pearson Education, Inc. 95% Confidence Interval for a Population Proportion For a population proportion, the margin of error for the 95% confidence interval is where is the sample proportion. The 95% confidence interval ranges from – margin of error to + margin of error We can write this confidence interval more formally as

46. Slide 8.3- 46 Copyright © 2009 Pearson Education, Inc. The Nielsen ratings for television use a random sample of households. A Nielsen survey results in an estimate that a women’s World Cup soccer game had 72.3% of the entire viewing audience. Assuming that the sample consists of n = 5,000 randomly selected households, find the margin of error and the 95% confidence interval for this estimate. Solution: EXAMPLE 2 TV Nielsen Ratings

47. Slide 8.3- 47 Copyright © 2009 Pearson Education, Inc. Solution: (cont.) EXAMPLE 2 TV Nielsen Ratings

48. Slide 8.3- 48 Copyright © 2009 Pearson Education, Inc. Choosing the Correct Sample Size In order to estimate a population proportion with a 95% degree of confidence and a specified margin of error of E, the size of the sample should be at least n = 1 E 2 Choosing Sample Size

49. Slide 8.3- 49 Copyright © 2009 Pearson Education, Inc. You plan a survey to estimate the proportion of students on your campus who carry a cell phone regularly. How many students should be in the sample if you want (with 95% confidence) a margin of error of no more than 4 percentage points? Solution: EXAMPLE 4 Minimum Sample Size for Survey