SPA 3e_ Teachers Edition _ Ch 6

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Answers continued X 18. (c) p^ = n , so X = p^ n; (0.33)(1500) = 495 and (0.37)(1500) = 555 495 − 525 (d) z = = −1.62; 18.47 555 − 525 z = = 1.62 18.47 P(495 # X # 555)= P(−1.62 ≤ Z # 1.62) = 0.9474 − 0.0526 = 0.8948 Using technology: Applet/normalcdf(lower: 495, upper:555, mean:525, SD:18.47) 5 0.8957. This answer matches the answer from the example that uses the sampling distribution of p^ , except for rounding. 19. (a) Shape: Both distributions are skewed to the right. Center: Drivers generally take longer to leave when someone is waiting for the space. Spread: There is more variability for the drivers with someone waiting. Outliers: There were no outliers for those with someone waiting, but there were two high outliers for those with no one waiting. (b) Not necessarily; the researchers merely observed what was happening, and they did not randomly assign the treatments of either having a person waiting or not to the drivers of the cars leaving the lot. 20. (a) Relative frequency 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Key Other Hazel Green Brown Blue 432 C H A P T E R 6 • Sampling Distributions (a) Write a few sentences comparing these distributions. (b) Can we conclude that a waiting car causes drivers to leave their spaces more slowly? Why or why not? 20. Those baby blues (2.1, 4.4) The two-way table summarizes information about eye color and gender in a random sample of 200 high school students. Gender Male Female Total Blue 21 29 50 Brown 35 40 75 Eye color green 14 21 35 Hazel 12 23 35 Other 2 3 5 Total 84 116 200 (a) Is there an association between eye color and gender in this group of students? Support your answer with an appropriate graphical summary of the data. (b) Select a student at random. Are the events “Student is male” and “Student has blue eyes” independent? Justify your answer. Lesson 6.5 The Sampling Distribution of a Sample Mean L e A r n i n g T A r g e T S d d Find the mean and standard deviation of the sampling distribution of a sample mean x and interpret the standard deviation. Use a normal distribution to calculate probabilities involving x when sampling from a normal population. When sample data are categorical, we often use the count or proportion of successes in the sample to make an inference about a population. When sample data are quantitative, we often use the sample mean x to estimate the mean m of a population. When we select random samples from a population, the value of x will vary from sample to sample. To understand how much x varies from m and what values of x are likely to happen by chance, we want to understand the sampling distribution of the sample mean x. Male Gender Female Yes, because the percentages for a given eye color are not the same for each gender. In other words, knowing a person’s gender helps us predict eye color. Males are more likely than females to have brown eyes, while females are more likely than males to have hazel or green eyes. (b) P(blue eyes | male) 5 21/84 5 0.25; P(blue eyes | female) 5 29/116 5 0.25. Because the probabilities are equal, the events “male” and “blue eyes” are independent. Knowing that a student is male does not change the probability that he has blue eyes. Starnes_3e_CH06_398-449_Final.indd 432 Learning Target Key The problems in the test bank are keyed to the learning targets using these numbers: d 6.5.1 d 6.5.2 BELL RINGER Thinking back to the “A penny for your thoughts?” activity for samples of 5 pennies, did every sample produce the same sample mean year? What is the name for this phenomenon? Teaching Tip Be picky with students about using the correct symbols! The sample mean is denoted x and the population mean is denoted m. Common Error This lesson and the next are about means, not proportions. Make sure students don’t use the symbols p and p^ when working with means. 18/08/16 5:02 PMStarnes_3e_CH0 432 C H A P T E R 6 • Sampling Distributions Starnes_3e_ATE_CH06_398-449_v3.indd 432 11/01/17 3:56 PM
L E S S O N 6.5 • The Sampling Distribution of a Sample Mean 433 DEFINITION Sampling distribution of the sample mean x The sampling distribution of the sample mean x describes the distribution of values taken by the sample mean x in all possible samples of the same size from the same population. When Mr. Ramirez’s class did the Penny for Your Thoughts activity at the beginning of the chapter, his students produced the “dotplot” in Figure 6.9 showing the simulated sampling distribution of x 5 the sample mean year of pennies in 50 samples of size n 5 5. x xx xxxxxx x x x 1990 1995 2000 2005 2010 2015 Sample mean year (n = 5) This distribution is slightly skewed to the left, with a mean of about 2002 and a standard deviation of about 5 years. By the end of Lesson 6.6, you should be able to anticipate the shape, center, and variability of distributions like this one without having to do a simulation. Center and Variability When we select random samples of size n from a population with mean m and standard deviation s, the value of x will vary from sample to sample. As with the sampling distribution of p^ , there are formulas that describe the center and variability of the sampling distribution of x. How to Calculate μ x and σ x Suppose that x is the mean of an SRS of size n drawn from a large population with mean m and standard deviation s. Then: • The mean of the sampling distribution of x is m x = m. • The standard deviation of the sampling distribution of x is s x = s "n . The behavior of x in repeated samples is much like that of the sample proportion p^ : • The sample mean x is an unbiased estimator of the population mean m. This is because the mean of the sampling distribution m x is equal to the mean of the population m. • The standard deviation of the sampling distribution of x describes the typical distance between the sample mean x and the population mean m. • The distribution of x is less variable for larger samples. This is indicated by the !n in the denominator of the standard deviation formula. • The formula for the standard deviation of the distribution of x requires that the observations be independent. In practice, we are safe assuming independence when we are sampling without replacement as long as the sample size is less than 10% of the population size. These facts about the mean and standard deviation of x are true no matter what shape the population distribution has. FigUre 6.9 Simulated sampling distribution of the sample mean year x in 50 samples of size n 5 5 from a population of pennies. Teaching Tip In Figure 6.9, ask students what the “dot” at 1997 represents. It is the sample mean/average year for one sample of 5 pennies. Teaching Tip This figure is a good opportunity to refer to the dotplots made by your students in the “A penny for your thoughts?” activity from Lesson 6.1. Compare the results from your class with those from Mr. Ramirez’s class. FYI The formulas for m x and s x are true for the sampling distribution of x no matter what shape it has. Teaching Tip: Differentiate There are many symbols in this section, which may be difficult for some students. These students may find it easier to read the bullet points by substituting the words “sample mean” for x and “population mean” for m. Teaching Tip This is consistent with previous definitions of standard deviation as the typical distance a value falls from the mean of a distribution. Lesson 6.5 18/08/16 5:02 PMStarnes_3e_CH06_398-449_Final.indd 433 18/08/16 5:02 PM L E S S O N 6.5 • The Sampling Distribution of a Sample Mean 433 Starnes_3e_ATE_CH06_398-449_v3.indd 433 11/01/17 3:56 PM
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