SPA 3e_ Teachers Edition _ Ch 6

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408 C H A P T E R 6 • Sampling Distributions 14. (a) Population: All guppies at the pet store. Parameter: m 5 the true mean length of the population. Sample: A random sample of 10 guppies. Statistic: The mean length of the sample; x 5 4.8 cm. (b) No; in the 200 simulated samples, 21 of the SRSs had a mean of 4.8 or less. Based on the simulation, P( x ≤ 4.8) = 21∙200 = 0.105. (c) No; because the probability from part (b) isn’t small, it is plausible that the true mean is m 5 5 and I got a sample mean of x 5 4.8 by chance alone. 15. (a) The distribution of heights for 16-year-old females is approximately normal with a mean of m 5 64 inches and standard deviation of s 5 2.5 inches. 56.5 59 61.5 64 66.5 Height (in.) 69 71.5 (b) Answers will vary. This is the distribution of one possible sample. d d d ddd d d ddd ddddddd d d 55 60 65 70 Height (in.) 16. (a) The distribution of measured weights for all scales is approximately normal with a mean of m 5 150 pounds and standard deviation of s 5 2 pounds. 144 146 148 150 152 Weight (lb) 154 156 (b) Answers will vary. This is the distribution of one possible sample. d d d d d dd dd d d d 146 147 148 149 150 151 152 153 154 155 Weight (lb) 17. (a) In 10 cases of taking a random sample of size n 5 50 from each high school, the difference in proportions of students with Internet access at home is 0%. This means the proportion of students with Internet access was the same for each high school in 10 pairs of simulated samples (b) Yes; in the 100 pairs of simulated samples, 0 of the pairs had a difference in proportions of 0.20 or higher. Based on the simulation, P(p^ N − p^ S ≥ 0.20) = 0∙100 = 0. (c) Yes; because the probability from part (b) is small, it is not plausible that the true difference in proportions is d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d ddd d dd d 4.6 4.8 5.0 5.2 5.4 Sample mean length (cm) (b) Would it be unusual to get a sample mean of x = 4.8 centimeters or less in a sample of size n 5 10 from this population? Explain. (c) Based on your answer to part (b), is there convincing evidence that the mean length of guppies at this store is less than 5 centimeters? Explain. 15. More tall girls Refer to Exercise 11. (a) Make a graph of the population distribution of heights for 16-year-old females. (b) Sketch a possible dotplot of the distribution of sample data for an SRS of size 20 from this population. 16. More bathroom scales Refer to Exercise 12. (a) Make a graph of the population distribution of weights, assuming the manufacturer’s claim is correct. (b) Sketch a possible dotplot of the distribution of sample data for an SRS of size 12 from this population. Extending the Concepts 17. Difference of proportions A school superintendent believes that the proportion of North High School students with Internet access at home is greater than the proportion of South High School students with Internet access at home. To investigate, she selects SRSs of size n 5 50 from each school and finds p^ N 5 46/50 5 0.92 and p^ S 5 36/50 5 0.72. To determine if a difference in proportions of 0.20 provides convincing evidence that North High School has a greater proportion of students with Internet access at home, we simulated two random samples of size n 5 50 from populations with the same proportion of students with Internet access. Then, we subtracted the sample proportions. Here are the results from repeating this process 100 times. d d d d d d d Starnes_3e_CH06_398-449_Final.indd 408 p N 2 p S 5 0 and the superintendent got a sample difference of proportion of p^ N − p^ S = 0.20 by chance alone. 18. (a) 300 C 25 = 1.95 × 10 36 different possible samples of 25 tomatoes. (b) Due to the extremely large number of possible samples, it is not practical to examine the complete sampling distribution of means for samples of size 25. 19. (a) z = x − m s ; 28,000 − 23,300 0.67 = ; s s = 7014.93 d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d –0.20 –0.10 0.00 0.10 0.20 d d d d d d d d d d Difference in proportion of students with Internet access at home d d d d d d d d d d d d d d d d d d d d d d d 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 are ten dots at 0. Explain what these dots represent. (b) Would it be unusual to get a difference in sample proportions of at least 0.20 when there is no difference in the population proportions? Explain. (c) Based on your answer to part (b), is there convincing evidence that North High School has a greater proportion of students with Internet access at home? Explain. Recycle and Review 18. Sampling tomatoes (4.8, 6.1) Zach runs a roadside stand during the summer, selling produce from his farm. On a single day in mid-August, he harvests 300 tomatoes. Suppose Zach wants to take a simple random sample of 25 tomatoes from the day’s pick to estimate mean weight. (a) How many possible sets of 25 tomatoes could be sampled from the 300 tomatoes in the day’s crop? (b) What does this say about the practicality of examining the complete sampling distribution of the sample mean for samples of size 25 from this population? 19. College debt (5.7) A report published by the Federal Reserve Bank of New York in 2012 reported the results of a nationwide study of college student debt. Researchers found that the average student loan balance per borrower is $23,300. They also reported that about one-quarter of borrowers owe more than $28,000. 4 (a) Assuming that the distribution of student loan balances is approximately normal, estimate the standard deviation of the distribution of student loan balances. (b) Assuming that the distribution of student loan balances is approximately normal, use your answer to part (a) to estimate the proportion of borrowers who owe more than $54,000. (c) In fact, the report states that about 10% of borrowers owe more than $54,000. What does this fact indicate about the shape of the distribution of student loan balances? (d) The report also states that the median student loan balance is $12,800. Does this fact support your conclusion in part (c)? Explain. 54,000 − 23,300 (b) z = ≈ 4.38; 7014.93 P(X ≥ 54,000)≈ P(Z ≥ 4.38) ≈ 0 Using technology: Applet/normalcdf(lower: 54000, upper:100000, mean:23300, SD:7014.93) 5 0.000006 (c) If the distribution of loan balances is approximately normal, then we would expect almost no one to have a balance that large. Because 10% of borrowers owe more than $54,000, we can conclude that the distribution of loan balances isn’t normal and is rightskewed. (d) Yes; because the mean ($23,300) is so much larger than the median ($12,800), we can conclude that the distribution of loan balances is skewed to the right. 18/08/16 4:59 PMStarnes_3e_CH0 408 C H A P T E R 6 • Sampling Distributions Starnes_3e_ATE_CH06_398-449_v3.indd 408 11/01/17 3:54 PM
ddd ddd dddd ddd ddd dd ddd ddd ddd Lesson 6.2 Sampling Distributions: center and variability L e A r n i n g T A r g e T S d Determine if a statistic is an unbiased estimator of a population parameter. d Describe the relationship between sample size and the variability of a statistic. Learning Target Key The problems in the test bank are keyed to the learning targets using these numbers: d 6.2.1 d 6.2.2 Lesson 6.2 AcT iviT y How many craft sticks are in the bag? In this activity, you will create a statistic for estimating the total number of craft sticks in a bag (N). The sticks are numbered 1, 2, 3, . . . , N. Near the end of the activity, your teacher will select a random sample of n 5 7 sticks and read the number on each stick to the class. The team that has the best estimate for the total number of sticks will win a prize. 1. Form teams of three or four students. As a team, spend about 10 minutes brainstorming different ways to estimate the total number of sticks. Try to come up with at least three different statistics. 2. Before your teacher provides the sample of sticks, use simulation to investigate the sampling distribution of each statistic. For the simulation, assume that there are N 5 100 sticks in the bag and that you will be selecting samples of size n 5 7. j Using your TI-83/84 calculator, select an SRS of size 7 using the command RandIntNoRep(lower: 1,upper:100,n:7). [With older OS, use the command RandInt(lower:1,upper:100,n:7) and verify that there are no repeated numbers. If there are repeats, press ENTER to get a new sample.] j For each sample, calculate the value of each of your three statistics. Unbiased Estimators j j graph these values on a set of dotplots like those shown here. Perform as many trials of the simulation as possible. Statistic 3 Statistic 2 Statistic 1 60 70 80 90 100 110 120 130 140 Estimated total 3. Based on the simulated sampling distributions, which of your statistics is likely to produce the best estimate? Discuss as a team. 4. Your teacher will now draw a sample of n 5 7 sticks from the bag. On a piece of paper, write the names of your group members, your group’s estimate for the number of sticks in the bag (a number), and the statistic you used to calculate your estimate (a formula). In the craft sticks activity, the goal was to estimate the maximum value in a population, with the assumption that the members of the population are numbered 1, 2, . . . , N. Two possible statistics that might be used to estimate N are the sample maximum (max) and twice the sample median (2 3 median). Assuming that the population has N 5 100 members and we use SRSs of size n 5 7, Figure 6.2 shows the simulated sampling distributions of the sample maximum and twice the sample median. 409 Bell Ringer Suppose you wish to estimate the average (mean) height of all students at your school by taking a random sample of students and calculating the average height of the sample. Would you expect the sample mean to be closer to the true average height from a sample of 4 students or 40 students? The best statistics are centered at 100 with low variability. A pre-made Fathom file is included in the Teacher’s Resource Materials. If you don’t have Fathom, the figure shows simulated sampling distributions for several commonly used statistics using 200 random samples. The statistics are • TwiceMean 5 2 · sample mean • TwiceMedian 5 2 · sample median • Max 5 sample maximum • MeanPlusMed 5 mean 1 median • SumQuartiles 5 Q 1 1 Q 3 • TwiceIQR 5 2 · IQR • MeanPlus2SD 5 sample mean 1 2 · sample standard deviation • Partition 5 (8/7) · sample maximum TwiceMean d d dd dd ddd d dd d d ddd dddd d dd d ddddddddd dddd d d dd ddddd d dd d dd d ddd d 18/08/16 4:59 PMStarnes_3e_CH06_398-449_Final.indd 409 Activity Overview Time: 40–50 minutes Materials: Graphing calculators or an Internetconnected device for each student or group of students, a prize for the winning team, and a population of craft sticks. We recommend using at least 100 sticks (but not exactly 100). Use an opaque container so students can’t use their eyes to estimate the total. Teaching Advice: The point of this activity is to illustrate the concepts of bias and variability of statistics. This is a long activity and should be done during an entire class period. Start by selecting one or two sticks to make the contents of the bag less abstract. Be patient in Step 1! Teams will struggle. 18/08/16 4:59 PM Let them keep at it. Once one statistic is proposed, usually more will follow. During Step 1, rotate around the room to give suggestions to teams that are struggling. Consider giving away one or more of the methods used later in the lesson, such as the sample maximum, twice the sample median, or twice the sample mean. These are all decent statistics but not the best. During Step 2, students can use their calculators or a random generator on the Internet (like random.org) to generate 7 integers without repeats. Groups should generate at least 20 samples. If you have the ability to simulate the sampling distributions for a number of statistics using Fathom or other software, consider offering another prize for the team with the best statistic. TwiceMedian Max MeanPlusMed SumQuartiles TwiceIQR MeanPlus2SD Partition d d dddd d d d ddd dd ddd ddd dd ddd ddd dddd dd d dd d d dddddddd ddd d dddd ddd d dd dd d dd dd d ddd d d dddd dd ddd ddddd d d d ddddddd ddddddddd d d dd ddddd d d d dd dd dddd dddd ddd dddd d ddd dd d dd d d d dd d d dd d ddddddd dd ddddddd d dd dd dd ddd dd d d ddd d dd d ddd dd d ddd d dd d dd dd ddd dddd ddddd d ddd dd d ddddd d ddd ddd dd d ddd d d dddd d d ddddd dddd d d dd ddd ddddddddd ddd d dd ddddd dddd ddddddddddd ddddd d 20 40 60 80 100 120 140 160 180 200 ddd d dddd dd dddd dd dd dddd d ddddd d ddd dd ddd d dd ddd dddd dddddddd d ddddd dddd dddddddd d d dd ddd d dd d ddd dd d dd d Answers: 1. Answers will vary by student group. 2. Dotplots will vary. 3. Answers will vary. The best statistics are centered at 100 with low variability. 4. Answers will vary by student group. dd d d d ddd dd dd dd d dd dd d d d ddd L E S S O N 6.2 • Sampling Distributions: Center and Variability 409 Starnes_3e_ATE_CH06_398-449_v3.indd 409 11/01/17 3:54 PM
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SPA 3e_ Teachers Edition _ Ch 6

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