2.43 MORE GRAPHS AND CHARTS

a z b z .125 .20 c z –z .668 2.0 d z .888 2.41

2.42 2.43

2.44 2.45 2.46 2.47 4 4 4 4 4 ` 4 ` 4 4 4 4 4 4 4 4 4 4 4 4 4 z z . z . z . z . X N , Y X Y . X Y X Y X Hint: s z x x s z x x s X x x 83

2.6 THE NORMAL DENSITY FUNCTION

Identify the values in the following diagrams of the standard normal distribu- tions interpolate as needed . Determine the quartiles of the standard normal distribution. Find the following: a. The area under the standard normal curve to the left of 35 b. The 35th percentile of the standard normal distribution c. The area under the standard normal curve to the left of 60 d. The 60th percentile of the standard normal distribution Find the following: a. The area under the standard normal curve to the left of 15 b. The 15th percentile of the standard normal distribution c. The area under the standard normal curve to the left of 99 d. The 99th percentile of the standard normal distribution If the variable has the 50 20 distribution, determine the distributions including the means and standard deviations of the following linear combina- tions. a. 10 3 b. 50 2 c. 25 2 d. 4 Show that standardized observations have sample mean 0 and sample standard deviation 1. [ Since is a constant, 0 see Exercise 1.21 of Chapter 1 . Also, . ] If is normally distributed with 40 and 3, find the following: a. The area under the normal curve to the left of 43 b. The area under the normal curve to the left of 45 1 1 2 2 2 1 1 2 2 2 2 2 m s n n i i i i n n i i i i

2.7 MORE GRAPHS AND CHARTS

2.48 2.49 2.50 2.51 2.52 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 x x x x x X x . x x x x x x x N , c c c c c N , c c c c c c c 84 CHAPTER 2 DESCRIBING PATTERNS IN DATA c. The area under the normal curve to the left of 38 d. The area under the normal curve to the right of 40 e. The area under the normal curve to the right of 36 f. The area under the normal curve over the interval 37 to 41 If is normally distributed with 15 and 6, find the following: a. The area under the normal curve to the left of 16 5 b. The area under the normal curve to the left of 7 c. The area under the normal curve to the right of 22 d. The area under the normal curve to the right of 11 e. The area under the normal curve over the interval 17 to 27 f. The area under the normal curve over the interval 1 to 19 For the 200 4 distribution, find the value such that a. The area under the normal curve to the left of is .8461. b. The area under the normal curve to the right of is .5897. c. The area under the normal curve to the left of is .0116. d. The area under the normal curve to the right of is .2297. For the 10 2 distribution, find the value such that a. The area under the normal curve to the left of is .7995. b. The area under the normal curve to the right of is .9429. c. The area under the normal curve over the interval to is .6826. d. The area under the normal curve over the interval to is .9544. Scores on a certain nationwide college entrance examination follow a normal distribution with a mean of 21 and a standard deviation of 3. Find the proportion of scores students a. Over 25 b. Under 20 c. Between 19 and 27 Refer to Exercise 2.51. a. If a school admits only students who score over 24, what proportion of the test takers would be eligible for admission? b. What limit would you set that makes 50 of the test takers eligible for admission? Round to the nearest integer. c. What would be the cutoff score if only the top 20 of the test takers are to be eligible? Round to the nearest integer. We cannot overemphasize the importance of graphing your data. If a feature you expect to see is not present in the plots, statistical analyses will, generally, be of little use. Some would argue that well-constructed graphs and charts convey virtually all 2 2 2 2 m s Flowchart Time Series Chart Pareto Chart Type Cause-and-Effect Histogram Measurement Time Measure Variable 1 Variable 2 Control Chart Scatter Diagram Measurement time UCL LCL x– p Figure 2.17 Seven Useful Charts The Deming Management Method. 85 p

2.7 MORE GRAPHS AND CHARTS

These charts are adapted from similar displays appearing in Walton, M. New York: Perigee Books, 1986. the information contained in the data. Figure 2.17 contains seven charts found to be particularly helpful in business and economics. We have already introduced most of these charts. The remaining charts will be introduced shortly. Creative graphics can highlight features in the data and even provide new insights. A classic example is the display by Charles Minard showing the ill-fated attempt of Kaunas 422,000 10,000 Niemen R. Berezina R. Smolensk Moscow 100,000 ° C –15 ° –30 ° –9 ° –21 ° –11 ° –20 ° –30 ° Dec. 6 Nov. 28 Nov. 14 Oct. 9 Temperature † p

2.8 STATISTICS IN CONTEXT