SPA 3e_ Teachers Edition _ Ch 6

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402 C H A P T E R 6 • Sampling Distributions Common Error The phrase “sampling distribution” sounds similar to “distribution of a sample,” but they mean very different things. In the “A penny for your thoughts?” activity, the distribution of a sample is distribution of year for the 5 (or 20) pennies in a student’s hand. The dotplots of the sample means and sample proportions created by the class are examples of sampling distributions. While some parameters and statistics have special symbols (such as p for the population proportion and p^ for the sample proportion), many parameters and statistics do not have their own symbol. To distinguish between a parameter and statistic, use descriptors such as “true” minimum and “sample” minimum as we did in the turkey example. Sampling Distributions In the Penny for Your Thoughts Activity, you encountered sampling variability— meaning that different random samples of the same size from the same population produce different values of a statistic. The statistics that come from these samples form a sampling distribution. DEFINITION Sampling distribution The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population. Alternate Example Disproportionate males? Sampling distributions PROBLEM: There are six employees in a small company, Atsuko, Bernadette, Carlos, Dandre, Easton, and Freddie. Atsuko and Bernadette are female and the others are male. List all 15 possible SRSs of size n 5 4, calculate the proportion of males for each sample, and display the sampling distribution of the sample proportion on a dotplot. SOLUTION: Sample 1: A, B, C, D p^ 5 0.50 Sample 2: A, B, C, E p^ 5 0.50 Sample 3: A, B, C, F p^ 5 0.50 Sample 4: A, B, D, E p^ 5 0.50 Sample 5: A, B, D, F p^ 5 0.50 Sample 6: A, B, E, F p^ 5 0.50 Sample 7: A, C, D, E p^ 5 0.75 Sample 8: A, C, D, F p^ 5 0.75 Sample 9: A, C, E, F p^ 5 0.75 Sample 10: A, D, E, F p^ 5 0.75 Sample 11: B, C, D, E p^ 5 0.75 Sample 12: B, C, D, F p^ 5 0.75 Sample 13: B, C, E, F p^ 5 0.75 Sample 14: B, D, E, F p^ 5 0.75 Sample 15: C, D, E, F p^ 5 1.00 a e XAMPLe Just how tall are their sons? Sampling distributions Remember that a distribution describes the possible values of a variable and how often these values occur. The easiest way to picture a distribution is with a graph, such as a dotplot or histogram. PROBLEM: John and Carol have four grown sons. Their heights (in inches) are 71, 75, 72, and 68. List all 6 possible SRSs of size n 5 2, calculate the mean height for each sample, and display the sampling distribution of the sample mean on a dotplot. SOLUTION: Sample 1: 71, 75 x 5 73 Sample 4: 75, 72 x 5 73.5 Sample 2: 71, 72 x 5 71.5 Sample 5: 75, 68 x 5 71.5 Sample 3: 71, 68 x 5 69.5 Sample 6: 72, 68 x 5 70 FigUre 6.1 Dotplot showing the sampling distribution of the sample range of height for SRSs of size n 5 2. Starnes_3e_CH06_398-449_Final.indd 402 69 70 71 72 73 74 Sample mean height (in.) FOR PRACTICE TRY EXERCISE 5. Every statistic has its own sampling distribution. For example, Figure 6.1 shows the sampling distribution of the sample range of height for SRSs of size n 5 2 from John and Carol’s four sons. Sample 1: 71, 75 sample range 5 4 Sample 4: 75, 72 sample range 5 3 Sample 2: 71, 72 sample range 5 1 Sample 5: 75, 68 sample range 5 7 Sample 3: 71, 68 sample range 5 3 Sample 6: 72, 68 sample range 5 4 d d d d d d 0 1 2 3 4 5 6 7 8 Sample range of height (in.) 18/08/16 4:58 PMStarnes_3e_CH0 d d d d d d d d 0.5 0.6 0.7 0.8 0.9 1.0 Sample proportion of men 402 C H A P T E R 6 • Sampling Distributions Starnes_3e_ATE_CH06_398-449_v3.indd 402 11/01/17 3:53 PM
L E S S O N 6.1 • What Is a Sampling Distribution? 403 Be specific when you use the word “distribution.” There are three different types of distributions in this setting: 1. The distribution of height in the population (the four heights): d d d d 67 68 69 70 71 72 73 74 75 76 Height (in.) 2. The distribution of height in a particular sample (two of the heights): d 67 68 69 70 71 72 73 74 75 76 Height (in.) 3. The sampling distribution of the sample range for all possible samples (the six sample ranges): d d d d d d 0 1 2 3 4 5 6 7 8 Sample range of height (in.) Notice that the first two distributions consist of heights (data values), while the third distribution consists of ranges (statistics). Lesson: Always use “the distribution of __” and never just “the distribution.” d cAutIOn ! Common Error Emphasize the difference in the three distributions shown here. It will be difficult, but important, for students to do this. Ask students to describe what the leftmost dot represents in each graph. In graph 1, it represents the height of a son (in the population of four sons) who is 68 inches tall. In graph 2, it represents the height of a son (in a sample of two sons) who is 71 inches tall. In graph 3, it represents the sample range of heights for the sample of the two sons who are 71 and 72 inches tall. Each dot in dotplot 3 represents a statistic from a sample, not a value from a single individual. Lesson 6.1 Using Sampling Distributions to Evaluate Claims Sampling distributions are the foundation for the methods of statistical inference you will learn about in Chapters 7–10. Knowing the sampling distribution of a statistic will help us know how much the statistic tends to vary from its corresponding parameter and what values of the statistic should be considered unusual. How long will we bead doing the homework? Evaluating a claim PROBLEM: At the beginning of class, Mrs. Chauvet shows her class a box filled with black and white beads. She claims that the proportion of black beads in the box is p 5 0.50. To determine the number of homework exercises she will assign that evening, she invites a student to select an SRS of n 5 30 beads from the box. The number of black beads selected will be the number of homework exercises assigned. When the student selects 19 black beads (p^ 5 19/30 5 0.63), the students groan and suggest that Mrs. Chauvet included more than 50% black beads in the box. To determine if a sample proportion of p^ 5 0.63 provides convincing evidence that Mrs. Chauvet cheated, the class simulated 100 SRSs of size n 5 30, assuming that she was telling the truth. That is, they sampled from a population with 50% black beads. For each sample, they recorded the sample proportion of black beads. The results of the simulation are shown on the next page. e XAMPLe © Monalyn Gracia/Corbis 18/08/16 4:58 PMStarnes_3e_CH06_398-449_Final.indd 403 Alternate Example What’s in the box? Evaluating a claim PROBLEM: At the end of class, Mr. Osters allows one student to select a ticket from a shoebox without looking. The tickets are labeled either “Homework pass” or “Try again.” Once a ticket is drawn, it is replaced for the next drawing and the tickets are mixed thoroughly. Mr. Osters claims that the proportion of homework passes in the shoebox is p 5 0.25. At the end of the first quarter, one student noted that only 6 students won in 50 drawings ( p^ 5 0.12). The students were suspicious that less than 25% of the tickets in the box are homework passes. 18/08/16 4:58 PM To determine if a sample proportion of p^ 5 0.12 provides convincing evidence that the true proportion of homework passes is less than 25%, the class simulated 100 SRSs of size n 5 50, assuming that 25% of the tickets were homework passes. For each sample, they recorded the sample proportion of homework passes. Here are the results of the simulation: d d dddddd d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Sample proportion of homework passes (a) There is one dot on the graph at p^ 5 0.38 Explain what this dot represents. (b) Would it be unusual to get a sample proportion of 0.12 or less in a sample of size 50 when p 5 0.25? Explain. (c) Based on your answer to part (b), is there convincing evidence that Mr. Osters lied about the contents of the shoebox? SOLUTION: (a) In one SRS of size n 5 50, 38% of the tickets were homework passes. (b) Yes; in the 100 trials of the simulation, only 2 of the SRSs included 12% or fewer homework passes. (c) Yes; because the probability from part (b) is small—only 0.02—it is not plausible that the proportion of homework passes in the shoebox is p 5 0.25 and the students got a sample proportion of p^ 5 0.12 by chance alone. L E S S O N 6.1 • What Is a Sampling Distribution? 403 Starnes_3e_ATE_CH06_398-449_v3.indd 403 11/01/17 3:53 PM
Page 2 and 3: 6 Sampling Distributions Please rea
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Page 6 and 7: Chapter 6 Resources SPA Applets hig
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Page 34 and 35: Answers continued 16. No; assuming
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Page 50 and 51: Activity answers continued 3. Yes,
Page 54 and 55: 9. (a) Because n = 50 ≥ 30, the s

SPA 3e_ Teachers Edition _ Ch 6

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