SPA 3e_ Teachers Edition _ Ch 6

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Activity answers continued 3. Yes, this confirms our observations in Step 2. 4. Students should paint their own population distribution. Yes, this is cool. The results from Steps 1 to 3 hold true for this new population (no matter what the population looks like)! 5. The sampling distribution of x has the same mean as the population distribution. As n increases, the variability in the sampling distribution decreases. The shape of the sampling distribution becomes more approximately normal as n increases. 440 C H A P T E R 6 • Sampling Distributions The Central Limit Theorem It is a remarkable fact that as the sample size increases, the sampling distribution of x changes shape: It looks less like that of the population and more like a normal distribution. This is true no matter what shape the population distribution has. This famous fact of probability theory is called the central limit theorem (sometimes abbreviated as CLT). DEFINITION central limit theorem (cLt) Draw an SRS of size n from any population with mean m and finite standard deviation s. The central limit theorem (CLT) says that when n is large, the sampling distribution of the sample mean x is approximately normal. How large a sample size n is needed for the sampling distribution of x to be close to normal depends on the population distribution. A larger sample size is required if the shape of the population distribution is far from normal. In that case, the sampling distribution of x will also be far from normal if the sample size is small. To use a normal distribution to calculate probabilities involving x, check the Normal/Large Sample condition. Teaching Tip It is hard to overstate the importance of the central limit theorem. Students should know this theorem by name. Make sure students understand that the CLT applies to sample means and refers to the shape (and only the shape) of the sampling distribution of x. Teaching Tip There is nothing magical about 30 as the boundary value for a “large” sample size. Some statisticians and authors recommend n 5 40 as the boundary. In truth, some populations need sample sizes much larger than 40 to have sampling distributions that are approximately normal. For our purposes, 30 is a reasonable value that works relatively well for most populations. Make sure that students understand that n ≥ 30 is just a guideline (like the Large Counts condition for proportions). a e XAMPLe A few more pennies for your thoughts? Sampling from a non-normal population DEFINITION normal/Large Sample condition The Normal/Large Sample condition says that the distribution of x will be approximately normal when either of the following is true: • The population distribution is approximately normal. This is true no matter what the sample size n is. • The sample size is large. If the population distribution is not normal, the sampling distribution of x will be approximately normal in most cases if n ≥ 30. PROBLEM: Mr. Ramirez’s class did the Penny for Your Thoughts Activity from the beginning of this chapter. The histogram shows the distribution of ages for the 2341 pennies in their collection. (a) Describe the shape of the sampling distribution of x for SRSs of size n 5 2 from the population of pennies. Justify your answer. (b) Describe the shape of the sampling distribution of x for SRSs of size n 5 50 from the population of pennies. Justify your answer. Frequency 140 120 100 80 60 40 20 0 1950 1960 1970 1980 1990 2000 2010 2020 Year Alternate Example Will you show me the money? Sampling from a non-normal population PROBLEM: Early in 2016, the opening weekend earnings from the top-earning movies of all time were reported by the website Box Office Mojo. The histogram shows the distribution of earnings, rounded to the nearest million dollars, for the top 1799 movies based on opening weekend gross income. Starnes_3e_CH06_398-449_Final.indd 440 Frequency 900 800 700 600 500 400 300 200 100 0 50 100 150 200 250 300 Opening weekend gross earnings ($ millions) (a) Describe the shape of the sampling distribution of x for SRSs of size n 5 60 from this population of movies. Justify your answer. (b) Describe the shape of the sampling distribution of x for SRSs of size n 5 10 from this population of movies. Justify your answer. SOLUTION: (a) Because n 5 60 ≥ 30, the sampling distribution of x will be approximately normal by the central limit theorem. (b) Because n 5 10 < 30, the sampling distribution of x will also be skewed to the right, but not quite as strongly as the population. 18/08/16 5:03 PMStarnes_3e_CH0 440 C H A P T E R 6 • Sampling Distributions Starnes_3e_ATE_CH06_398-449_v3.indd 440 11/01/17 3:58 PM
L E S S O N 6.6 • The Central Limit Theorem 441 SOLUTION: (a) Because n 5 2 < 30, the sampling distribution of x will be skewed to the left, but not quite as strongly as the population. (b) Because n 5 50 ≥ 30, the sampling distribution of x will be approximately normal by the central limit theorem. FOR PRACTICE TRY EXERCISE 1. Lesson 6.6 The dotplots in Figure 6.11 show the simulated sampling distributions of the sample mean for (a) 500 SRSs of size n 5 2 and (b) 500 SRSs of size n 5 50. 1970 d d d d d d d d d d d d d d d d d d d d d d d 1980 1990 2000 2010 Sample mean (n = 2) 2020 d d d d d d d d d d d d d d ddddddddddddddddddddddddddddddddddddddddddddd d ddddddddddddddddddddddddddddddddddddddddddddddddd d ddddddddddddddddddddddddddddddddddddddddddddddddddddddd d 1998 2000 2002 2004 2006 Sample mean (n = 50) As expected, the simulated sampling distribution of x for SRSs of size n 5 2 is skewed to the left and the simulated sampling distribution of x for SRSs of size n 5 50 is approximately normal—thanks to the central limit theorem. Probabilities Involving x Using the central limit theorem, we can do probability calculations involving x even when the population is non-normal. Mean texts? Probabilities involving x PROBLEM: Suppose that the number of texts sent during a typical day by the population of students at a large high school follows a right-skewed distribution with a mean of 45 and a standard deviation of 35. How likely is it that a random sample of 100 students will average at least 50 texts per day? SOLUTION: • Mean: m- x 5 45 texts • SD: s x - = 35 5 3.5 texts "100 • Shape: Approximately normal by the CLT because n 5 100 ≥ 30 d d d d d d d FigUre 6.11 Simulated sampling distributions of the sample mean age for (a) 500 SRSs of size n 5 2 and (b) 500 SRSs of size n 5 50 from a population of pennies. e XAMPLe Let x = the sample mean number of texts. To find P(x ≥ 50), we have to know the mean, standard deviation, and shape of the sampling distribution of x. Recall that m x -= m and s– x = s "n . Common Error Remind your students that the CLT only addresses the shape of the sampling distribution of x. It doesn’t tell us about the center or variability of the sampling distribution. Alternate Example Will you show me more money? Probabilities involving x PROBLEM: The opening weekend earnings for the top 1799 movies of all time have a distribution that is strongly rightskewed with a mean of $26.2 million and standard deviation of $22.9 million. What is the probability that a random sample of 50 of these movies will have average opening weekend earnings of less than $19 million? SOLUTION: • Mean: m x = $26.2 million • SD: s x = $22.9 = $3.24 million "50 • Shape: Approximately normal by the CLT because n 5 50 ≥ 30 18/08/16 5:03 PMStarnes_3e_CH06_398-449_Final.indd 441 18/08/16 5:04 PM 19 16.48 19.72 22.96 26.20 29.44 32.68 35.92 Sample mean opening weekend gross earnings ($ millions) 19 − 26.2 Using Table A: z = = −2.22 3.24 and P(Z < 22.22) 5 0.0132 Using technology: Applet/normalcdf (lower: 2100000, upper: 19, mean: 26.2, SD: 3.24) 5 0.0131 L E S S O N 6.6 • The Central Limit Theorem 441 Starnes_3e_ATE_CH06_398-449_v3.indd 441 11/01/17 3:58 PM
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Page 4 and 5: Promoting Good Habits and Skills Ch
Page 6 and 7: Chapter 6 Resources SPA Applets hig
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SPA 3e_ Teachers Edition _ Ch 6

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