SPA 3e_ Teachers Edition _ Ch 6

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406 C H A P T E R 6 • Sampling Distributions Exercises 5–8 refer to the following population of 2 Teaching Tip male students and 3 female students, along with their quiz scores: Exercises 5–8 are very important because Abigail 10 Bobby 5 Carlos 10 DeAnna 7 Emily 9 students can create (and see) an entire 5. Sample means List all 10 possible SRSs of size sampling distribution. Don’t skip these pg 402 n 5 2, calculate the mean quiz score for each sample, and display the sampling distribution of the sample exercises! mean on a dotplot. 6. Sample ranges List all 10 possible SRSs of size 5. n 5 3, calculate the range of quiz scores for each sample, and display the sampling distribution of Sample #1: Abigail (10), Bobby (5) x 5 7.5 the sample range on a dotplot. Sample #2: Abigail (10), Carlos (10) x 5 10 7. Sample proportions List all 10 possible SRSs of size n 5 2, calculate the proportion of females for each Sample #3: Abigail (10), DeAnna (7) x 5 8.5 sample, and display the sampling distribution of Sample #4: Abigail (10), Emily (9) x 5 9.5 the sample proportion on a dotplot. 8. Sample medians List all 10 possible SRSs of size Sample #5: Bobby (5), Carlos (10) x 5 7.5 n 5 3, calculate the median quiz score for each Sample #6: Bobby (5), DeAnna (7) x 5 6 sample, and display the sampling distribution of the sample median on a dotplot. Sample #7: Bobby (5), Emily (9) x 5 7 9. Who does their homework? A school newspaper Sample #8: Carlos (10), DeAnna (7) x 5 8.5 pg 403 article claims that 60% of the students at a large high school completed their assigned homework Sample #9: Carlos (10), Emily (9) x 5 9.5 last week. Some statistics students want to investigate if this claim is true, so they choose an SRS of Sample #10: DeAnna (7), Emily (9) x 5 8 100 students from the school to interview. When d d d they found that only 45 of the 100 students completed their assigned homework last week, they d d d d d d d 6 6.5 7 7.5 8 8.5 9 9.5 10 suspected that the proportion of all students who Sample mean quiz score completed their assigned homework last week is less than the 60% claimed by the newspaper. 6. To determine if a sample proportion of p^ 5 0.45 provides convincing evidence that the true proportion is less than p 5 0.60, the class simulated 250 Sample #1: Abigail (10), range 5 5 Bobby (5), Carlos (10) SRSs of size n 5 100 from a population in which p 5 0.60. Here are the results of the simulation. Sample #2: Abigail (10), range 5 5 d d Bobby (5), DeAnna (7) d Sample #3: Abigail (10), range 5 5 d d d d d d d d d Bobby (5), Emily (9) dddd d d d d d d d d ddd 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 Sample #4: Abigail (10), range 5 3 d d d d d d Carlos (10), DeAnna (7) d d d d d d dddd 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 Sample #5: Abigail (10), range 5 1 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 Carlos (10), Emily (9) dd dddddddddd dddddddddddddddd d dddddd d ddddddd d d d d d d d d d d d d d d d d d d d d d d dddddd d d dd d d d Sample #6: Abigail (10), range 5 3 0.45 0.50 0.55 0.60 0.65 0.70 0.75 DeAnna (7), Emily (9) Sample proportion of students who completed homework Sample #7: Bobby (5), range 5 5 Carlos (10), DeAnna (7) (a) There is one dot on the graph at 0.73. Explain what this dot represents. Sample #8: Bobby (5), range 5 5 (b) Would it be surprising to get a sample proportion of 0.45 or less in an SRS of size 100 when Carlos (10), Emily (9) Sample #9: Bobby (5), range 5 4 p 5 0.60? Explain. DeAnna (7), Emily (9) Sample #10: Carlos (10), range 5 3 DeAnna (7), Emily (9) d d dd d d d d 1 1.5 2 2.5 3 3.5 4 4.5 5 Starnes_3e_CH06_398-449_Final.indd 406 Sample range of quiz score 8. Sample #1: Abigail (10), median 5 10 7. Bobby (5), Carlos (10) Sample #1: Abigail, Bobby p^ 5 0.50 Sample #2: Abigail (10), median 5 7 Sample #2: Abigail, Carlos Bobby (5), DeAnna (7) p^ 5 0.50 Sample #3: Abigail (10), median 5 9 Sample #3: Abigail, DeAnna p^ 5 1 Bobby (5), Emily (9) Sample #4: Abigail, Emily p^ 5 1 Sample #4: Abigail (10), median 5 10 Sample #5: Bobby, Carlos p^ 5 0 Carlos (10), DeAnna (7) Sample #5: Abigail (10), median 5 10 Sample #6: Bobby, DeAnna p^ 5 0.50 Carlos (10), Emily (9) Sample #7: Bobby, Emily p^ 5 0.50 Sample #6: Abigail (10), median 5 9 Sample #8: Carlos, DeAnna p^ 5 0.50 DeAnna (7), Emily (9) Sample #9: Carlos, Emily Sample #7: Bobby (5), median 5 7 p^ 5 0.50 Carlos (10), DeAnna (7) Sample #10: DeAnna, Emily p^ 5 1 Sample #8: Bobby (5), median 5 9 Carlos (10), Emily (9) Sample #9: Bobby (5), median 5 7 d d ddddd d dd DeAnna (7), Emily (9) 0 0.5 1 Sample #10: Carlos (10), median 5 9 Sample proportion of females DeAnna (7), Emily (9) (c) Based on your answer to part (b), is there convincing evidence that the proportion of all students who completed their assigned homework last week is less than p 5 0.60? Explain. 10. First-serve percentage One important aspect of a tennis player’s effectiveness is her first-serve percentage—the proportion of the time the first of her two attempts to serve the ball to her opponent is successful. For her first three years on the tennis team, Shruti’s first-serve percentage is 53%. Hoping to improve, Shruti works over the summer with a coach who specializes in serves. In her first match of the next season, Shruti’s first serve is successful 42 times in 60 attempts, a first-serve percentage of 70%. Suppose we treat Shruti’s first 60 attempts as an SRS of her serves after working with the new coach. To determine if a sample proportion of p^ 5 0.70 provides convincing evidence that the true proportion is greater than p 5 0.53, we simulate 200 SRSs of size n 5 60 from a population in which p 5 0.53. Here are the results of the simulation. d d d d d d d d d d d d d d d d d d d d d d d ddddddddd d d d d d d d d d d d d d d dd d d d d d d d d d d d d d d d d d d d d d d ddd d d d d d d d d d d d d d d d d d d d dd d d d d d d d d d d ddddddddddddddddd 0.37 0.40 0.43 0.47 0.50 0.53 0.57 0.60 0.63 0.67 0.70 Sample proportion of successful first serves d d d d d d d 7 7.5 8 8.5 9 9.5 10 Sample median of quiz scores 9. (a) In one SRS of size n 5 100, 73% of the students did all their assigned homework. (b) Yes; in the 250 simulated samples, 0 of the SRSs had a sample proportion of 0.45 or lower. Based on the simulation, P( p^ ≤ 0.45) = 0∙250 = 0. (c) Yes; because the probability from part (b) is small, it is not plausible that the true proportion is p 5 0.60 and the statistics students got a sample proportion of p^ 5 0.45 by chance alone. 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 (a) There is one dot on the graph at 0.37. Explain what this dot represents. (b) Would it be surprising to get a sample proportion of 0.70 or more in an SRS of size 60 when p 5 0.53? Explain. (c) Based on your answer to part (b), is there convincing evidence that Shruti’s first-serve percentage has improved since working with the new coach? Explain. 11. Are we taller? According to the National Center for Health Statistics, the distribution of heights for 16-year-old females is modeled well by a normal distribution with mean m 5 64 inches and standard deviation s 5 2.5 inches. To see if this distribution applies at their high school, a statistics class takes an SRS of 20 of the 300 16-year-old females at the school and measures their heights. When they calculate a sample mean of 64.7 inches, they wonder if the population of 16-year-old girls at their school has a mean height greater than 64 inches. To determine if a sample mean of x 5 64.7 inches provides convincing evidence that the average height of 16-year-old girls at the school is taller Answers 10–11 are on page 407 18/08/16 4:59 PMStarnes_3e_CH0 406 C H A P T E R 6 • Sampling Distributions Starnes_3e_ATE_CH06_398-449_v3.indd 406 11/01/17 3:53 PM
L E S S O N 6.1 • What Is a Sampling Distribution? 407 than 64 inches, the class simulated 200 SRSs of size n 5 20 from a normal population with mean m 5 64 inches and standard deviation s 5 2.5 inches. Here are the results of the simulation. 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 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 d d d d d d d d d d d dddd 62.5 63.0 63.5 64.0 64.5 65.0 65.5 66.0 Sample mean height (in.) (a) There is one dot on the graph at 62.5. Explain what this dot represents. (b) Would it be unusual to get a sample mean of 64.7 or more in a sample of size 20 when m 5 64? Explain. (c) Based on your answer to part (b), is there convincing evidence that the mean height of the population of 16-year-old girls at this school is greater than 64 inches? Explain. 12. Relying on bathroom scales A manufacturer of bathroom scales says that when a 150-pound weight is placed on a scale produced in the factory, the weight indicated by the scale is normally distributed with a mean of 150 pounds and a standard deviation of 2 pounds. A consumeradvocacy group acquires an SRS of 12 scales from the manufacturer and places a 150-pound weight on each one. The group gets a mean weight of 149.1 pounds, which makes them suspect that the scales underestimate the true weight. To test this, they use a computer to simulate 200 samples of 12 scales from a population with a mean of 150 pounds and standard deviation of 2 pounds. Here is a dotplot of the means from these 200 samples. 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 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 d d d d d d d d d d d d dd d d 148.0 148.5 149.0 149.5 150.0 150.5 151.0 151.5 Sample mean weight (lb) (a) There is one dot on the graph at 151.2. Explain what this dot represents. d (b) Would it be unusual to get a sample mean of 149.1 or less in a sample of size 12 when m 5 150? Explain. (c) Based on your answer to part (b), is there convincing evidence that the scales produced by this manufacturer underestimate true weight? Explain. Applying the Concepts 13. Instant winners A fast-food restaurant promotes certain food items by giving a game piece with each item. Advertisements proclaim that “25% of the game pieces are Instant Winners!” To test this claim, a frequent diner collects 20 game pieces and gets only 3 instant winners. (a) Identify the population, the parameter, the sample, and the statistic in this context. Suppose the advertisements are correct and p 5 0.25. The dotplot shows the distribution of the sample proportion of instant winners in 100 simulated SRSs of size n 5 20. 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 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Sample proportion of instant winners (b) Would it be unusual to get a sample proportion of p^ 5 3/20 5 0.15 or less in a sample of size n 5 20 when p 5 0.25? Explain. (c) Based on your answer to part (b), is there convincing evidence that fewer than 25% of all game pieces are instant winners? Explain. 14. Puny guppies? A large pet store that specializes in tropical fish has several thousand guppies. The store claims that the lengths of its guppies are approximately normally distributed with a mean of 5 centimeters and a standard deviation of 0.5 centimeter. You come to the store and buy 10 randomly selected guppies and find that the mean length of your 10 guppies is only 4.8 centimeters. (a) Identify the population, the parameter, the sample, and the statistic in this context. Suppose the store’s description of the lengths of its guppies is true. The dotplot on the next page shows the distribution of sample means from 200 simulated SRSs of size n 5 10 from a normally distributed population with m 5 5 centimeters and s 5 0.5 centimeter. d d d d d d d d d d d d d d d d d d d d d d d 11. (a) In one SRS of size n 5 20, the mean height was 62.5 inches. (b) No, in the 200 simulated samples, 23 of the SRSs had a mean of 64.7 or more. Based on the simulation, P( x ≥ 64.7) = 23∙200 = 0.115. (c) No; because the probability from part (b) isn’t small, it is plausible that the true mean is m 5 64 and the class got a sample mean of x 5 64.7 by chance alone. 12. (a) In one SRS of size n 5 12, the mean weight was 151.2 pounds. (b) No; in the 200 simulated samples, 22 of the SRSs had a mean of 149.1 or less. Based on the simulation, P( x ≤ 149.1) = 22∙200 = 0.11. (c) No; because the probability from part (b) isn’t small, it is plausible that the true mean is m 5 150 and the group got a sample mean of x 5 149.1 by chance alone. 13. (a) Population: All game pieces. Parameter: p 5 the true proportion of the population that are instant winners. Sample: The 20 game pieces collected by the frequent diner. Statistic: The proportion of the sample that are instant winners; p^ 5 3/20 5 0.15. (b) No; in the 100 simulated samples, 18 of the SRSs had a sample proportion of 0.15 or lower. Based on the simulation, P( p^ ≤ 0.15) = 18∙100 = 0.18. (c) No; because the probability from part (b) isn’t small, it is plausible that the true proportion is p 5 0.25 and the frequent diner got a sample proportion of p^ 5 0.15 by chance alone. Lesson 6.1 18/08/16 4:59 PMStarnes_3e_CH06_398-449_Final.indd 407 Answers continued 10. (a) In one SRS of size n 5 60, 36.7% of the first serves were successful. (b) Yes; in the 200 simulated samples, only 1 of the SRSs had a sample proportion of 0.70 or higher. Based on the simulation, P( p^ ≥ 0.70) = 1∙200 = 0.005. (c) Yes; because the probability from part (b) is small, it is not plausible that the true proportion is still p 5 0.53 and the player got a sample proportion of p^ 5 0.70 by chance alone. 18/08/16 4:59 PM L E S S O N 6.1 • What Is a Sampling Distribution? 407 Starnes_3e_ATE_CH06_398-449_v3.indd 407 11/01/17 3:53 PM
Page 2 and 3: 6 Sampling Distributions Please rea
Page 4 and 5: Promoting Good Habits and Skills Ch
Page 6 and 7: Chapter 6 Resources SPA Applets hig
Page 8 and 9: PD Chapter 6 Overview Watch the cha
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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

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