SPA 3e_ Teachers Edition _ Ch 6

More documents

Recommendations

Info

416 C H A P T E R 6 • Sampling Distributions 15. (a) Statistics (ii) and (iii) both appear to be unbiased because the mean of each sampling distribution is very close to the value of the population parameter. (b) Statistic (ii); while both statistics (ii) and (iii) are unbiased, statistic (ii) has lower variability. 16. (a) 10 + 5 + 10 + 7 + 9 41 m = = 5 5 = 8.2 (b) Sample #1: Abigail (10), x 5 7.5 Bobby (5) Sample #2: Abigail (10), x 5 10 Carlos (10) Sample #3: Abigail (10), x 5 8.5 DeAnna (7) Sample #4: Abigail (10), x 5 9.5 Emily (9) Sample #5: Bobby (5), x 5 7.5 Carlos (10) Sample #6: Bobby (5), x 5 6 DeAnna (7) Sample #7: Bobby (5), x 5 7 Emily (9) Sample #8: Carlos (10), x 5 8.5 DeAnna (7) Sample #9: Carlos (10), x 5 9.5 Emily (9) Sample #10: DeAnna (7), x 5 8 Emily (9) (c) d d d d d d d d d 6 6.5 7 7.5 8 8.5 9 9.5 10 Sample mean quiz score m x = 7.5 + 10 + 8.5 + 9.5 + 7.5 + 6 + 7 + 8.5 + 9.5 + 8 10 = 82 10 = 8.2. Yes, the sample mean is an unbiased estimator of the population mean. The mean of the sampling distribution is equal to 8.2, which is the value of the population mean. (a) Which statistics are unbiased estimators? Justify your answer. (b) Which statistic does the best job of estimating the parameter? Explain. Extending the Concepts 16. More about means In the Exercises for Lesson 6.1, you were introduced to the following population of 2 male students and 3 female students, along with their quiz scores: Abigail 10 Bobby 5 Carlos 10 DeAnna 7 Emily 9 (a) Calculate the mean quiz score for the entire population. (b) List all 10 possible SRSs of size n 5 2, calculate the mean quiz score for each sample, and display the sampling distribution of the sample mean in a dotplot. (c) Calculate the mean of the sampling distribution from part (b). Is the sample mean an unbiased estimator of the population mean? Explain. 17. More about proportions In the Exercises for Lesson 6.1, you were introduced to the following population of 2 male students and 3 female students, along with their quiz scores: Abigail 10 Bobby 5 Carlos 10 DeAnna 7 Emily 9 (a) Calculate the proportion of females in the entire population. (b) List all 10 possible SRSs of size n 5 2, calculate the proportion of females for each sample, and display the sampling distribution of the sample proportion in a dotplot. (c) Calculate the mean of the sampling distribution from part (b). Is the sample proportion an unbiased estimator of the population proportion? Explain. Recycle and Review 18. Students and housing (4.3, 4.4) There are 104 students in Professor Negroponte’s statistics class, 49 males and 55 females. Sixty of the students live in the dorms and the rest live off campus. Twenty of the males live off-campus. Choose a student at random from this class. Let Event M 5 the student is male and Event D 5 the student lives in the dorms. (a) Construct a Venn diagram to represent the outcomes of this chance process using the events M and D. (b) Find each of the following probabilities and interpret them in context. (i) P(M c D) (ii) P(M C d D) (iii) P(D k M) 19. Students and homework (5.3, 5.4) Refer to Exercise 18. At the beginning of each day that Professor Negroponte’s class meets, he randomly selects a member of the class to present the solution to a homework problem. Suppose the class meets 40 times during the semester and the selections are made with replacement. Let X 5 the number of times a female student is selected to present a solution. (a) Is X a binomial random variable? Justify your answer. (b) Calculate the mean and standard deviation of X. (c) For the first 10 meetings of the class, Professor Negroponte selects only 1 female student to solve a problem. Is there convincing evidence that his selection process is not really random? Support your answer with an appropriate probability calculation. 17. (a) p = 3 5 = 0.6 (b) Sample #1: Abigail, Bobby p^ 5 0.5 Sample #2: Abigail, Carlos p^ 5 0.5 Sample #3: Abigail, DeAnna p^ 5 1 Sample #4: Abigail, Emily p^ 5 1 Sample #5: Bobby, Carlos p^ 5 0 Sample #6: Bobby, DeAnna p^ 5 0.5 Sample #7: Bobby, Emily p^ 5 0.5 Sample #8: Carlos, DeAnna p^ 5 0.5 Sample #9: Carlos, Emily p^ 5 0.5 Sample #10: DeAnna, Emily p^ 5 1 d d d d dd d d 0 0.5 1 Sample proportion of female Starnes_3e_CH06_398-449_Final.indd 416 (c) 0.5 + 0.5 + 1 + 1 + 0 + 0.5 + 0.5 + 0.5 + 0.5 + 1 m p^ = = 6 10 10 = 0.6. Yes, the sample proportion is an unbiased estimator of the population proportion. The mean of the sampling distribution is equal to 0.6, which is the value of the population proportion. 18. (a) Male 20 29 Dorms 31 24 20 + 29 + 31 (b) (i) P(M c D) = P(M or D) = 104 = 80 = 0.769. There is about a 77% chance 104 that a randomly selected student is a male or lives in the dorm. (ii) P(M C d D) = P(M C 31 and D)= 104 = 0.298. There is about a 30% chance that a randomly selected student is not a male and lives in the dorm. P(D and M) (iii) P(D 0 M) = = 29∙104 P(M) 49∙104 = 29 49 = 0.592. There is about a 59% chance that a randomly selected student lives in the dorm, given that the student is a male. Answer 19 is on page 417 18/08/16 5:00 PMStarnes_3e_CH0 416 C H A P T E R 6 • Sampling Distributions Starnes_3e_ATE_CH06_398-449_v3.indd 416 11/01/17 3:54 PM
18/08/16 5:00 PMStarnes_3e_CH06_398-449_Final.indd 417 Lesson 6.3 The Sampling Distribution of a Sample count (The normal Approximation to the Binomial) L e A r n i n g T A r g e T S d Calculate the mean and the standard deviation of the sampling distribution of a sample count and interpret the standard deviation. d Determine if the sampling distribution of a sample count is approximately normal. d If appropriate, use the normal approximation to the binomial distribution to calculate probabilities involving a sample count. In many cases, we are interested in the number of successes X in a random sample from some population. For example, X 5 the number of defective flash drives in a random sample of 10 flash drives or X 5 the number of Democrats in a random sample of 1000 registered voters. To do probability calculations involving X, we want an understanding of the sampling distribution of the sample count X. DEFINITION Sampling distribution of the sample count X The sampling distribution of the sample count X describes the distribution of values taken by the sample count X in all possible samples of the same size from the same population. The sampling distribution of X is closely related to the binomial distributions that you learned about in Lessons 5.3 and 5.4. Suppose that a supplier inspects an SRS of 10 flash drives from a shipment of 10,000 flash drives in which 200 are defective. Let X 5 the number of bad flash drives in the sample. This is not quite a binomial setting. Because we are sampling without replacement, the independence condition is violated. The conditional probability that the second flash drive chosen is bad changes when we know whether the first is good or bad: P(second is bad | first is good) 5 200/9999 5 0.0200 but P(second is bad | first is bad) 5 199/9999 5 0.0199. These probabilities are very close because removing 1 flash drive from a shipment of 10,000 changes the makeup of the remaining 9999 flash drives very little. The distribution of X is very close to the binomial distribution with n 5 10 and p 5 0.02. Answers continued 19. (a) Yes. Binary? “Success” 5 female is selected. “Failure” 5 male is selected. Independent? Knowing whether or not one randomly selected student is a female tells you nothing about whether or not another randomly selected student is a female. Number? n 5 40. Same probability? p = 55 104 = 0.529 (b) m X = np = 40(0.529) = 21.16 s X = "np(1 − p) = "40(0.529)(1 − 0.529) = "40(0.529)(0.471) = "9.97 = 3.16 (c) P(X = 0) = 10 C 0 (0.529) 0 (1 − 0.529) 10 = 1(0.529) 0 (0.471) 10 = 0.0005 P(X = 1) = 10C 1 (0.529) 1 (1 − 0.529) 9 = 10(0.529) 1 (0.471) 9 = 0.006 417 18/08/16 5:00 PM P(X ≤ 1) = P(X = 0) + P(X = 1) = 0.0005 + 0.006 = 0.0065 If the professor were to randomly choose students for the first 10 meetings, there is less than a 1% chance that he would select 1 female or fewer purely by chance. Because this is unlikely, we have convincing evidence that his selection process is not really random. Teaching Tip To save time, Lessons 6.2 and/or 6.3 can be skipped without losing much continuity in future chapters. However, the other lessons in this chapter are crucial to understanding much of the remainder of the course. Learning Target Key The problems in the test bank are keyed to the learning targets using these numbers: d 6.3.1 d 6.3.2 d 6.3.3 BELL RINGER According to the manufacturer, the true proportion of blue M&M’S® milk chocolate candies is 0.24. If Mrs. Gallas takes a random sample of 50 candies, how many should she expect to be blue? If she repeatedly takes random samples of 50 candies, will she always get the same number of blue candies? Why or why not? FYI When sampling with replacement from a finite population or sampling from an infinite population, the distribution of the sample count is exactly binomial. However, these types of sampling methods are not common in practice. It is far more common to sample without replacement from a finite population. In this case, the sample count is usually approximately binomial. Lesson 6.3 L E S S O N 6.3 • The Sampling Distribution of a Sample Count 417 Starnes_3e_ATE_CH06_398-449_v3.indd 417 11/01/17 3:54 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
Page 10 and 11: PD LESSONS 6.1-6.3 Overview Watch t
Page 12 and 13: 402 C H A P T E R 6 • Sampling Di
Page 22 and 23: dd dd d d dddddd 412 C H A P T E R
Page 34 and 35: Answers continued 16. No; assuming
Page 42 and 43: Answers continued X 18. (c) p^ = n
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?