272 CHAPTER 6 Normal Probability Distributions 37. Curving Test Scores A professor gives a test and the scores are normally distributed with a mean of 60 and a standard deviation of 12. She plans to curve the scores. a. If she curves by adding 15 to each grade, what is the new mean and standard deviation? b. Is it fair to curve by adding 15 to each grade? Why or why not? c. If the grades are curved so that grades of B are given to scores above the bottom 70% and below the top 10%, find the numerical limits for a grade of B. d. Which method of curving the grades is fairer: adding 15 to each original score or using a scheme like the one given in part (c)? Explain. 38. Outliers For the purposes of constructing modified boxplots as described in Section 3-3, outliers are defined as data values that are above Q3 by an amount greater than 1.5 * IQR or below Q1 by an amount greater than 1.5 * IQR, where IQR is the interquartile range. Using this definition of outliers, find the probability that when a value is randomly selected from a normal distribution, it is an outlier. 6-2 Beyond the Basics Key Concept We now consider the concept of a sampling distribution of a statistic. Instead of working with values from the original population, we want to focus on the values of statistics (such as sample proportions or sample means) obtained from the population. Figure 6-16 shows the key points that we need to know, so try really, really hard to understand the story that Figure 6-16 tells. A Statistics Story Among the population of all adults, exactly 70% do not feel comfortable in a self-driving vehicle (the author just knows this). In a TE Connectivity survey of 1000 adults, 69% said that they did not feel comfortable in a self-driving vehicle. Empowered by visions of multitudes of driverless cars, 50,000 people became so enthusiastic that they each conducted their own individual survey of 1000 randomly selected adults on the same topic. Each of these 50,000 newbie surveyors reported the percentage that they found, with results such as 68%, 72%, 70%. The author obtained each of the 50,000 sample percentages, he changed them to proportions, and then he constructed the histogram shown in Figure 6-17. Notice anything about the shape of the histogram? It’s normal (unlike the 50,000 newbie surveyors). Notice anything about the mean of the sample proportions? They are centered about the value of 0.70, which happens to be the population proportion. Moral: When samples of the same size are taken from the same population, the following two properties apply: 1. Sample proportions tend to be normally distributed. 2. The mean of sample proportions is the same as the population mean. The implications of the preceding properties will be extensive in the chapters that follow. What a happy ending! 6-3 Sampling Distributions and Estimators
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