296 CHAPTER 6 Normal Probability Distributions PART 2 Manual Construction of Normal Quantile Plots The following is a relatively simple procedure for manually constructing a normal quantile plot, and it is the same procedure used by Statdisk and the TI-83>84 Plus calculator. Some statistical packages use various other approaches, but the interpretation of the graph is essentially the same. Manual Construction of a Normal Quantile Plot Step 1: First sort the data by arranging the values in order from lowest to highest. Step 2: With a sample of size n, each value represents a proportion of 1>n of the sample. Using the known sample size n, find the values of 1 2n , 3 2n , 5 2n , and so on, until you get n values. These values are the cumulative areas to the left of the corresponding sample values. Step 3: Use the standard normal distribution (software or a calculator or Table A-2) to find the z scores corresponding to the cumulative left areas found in Step 2. (These are the z scores that are expected from a normally distributed sample.) Step 4: Match the original sorted data values with their corresponding z scores found in Step 3, then plot the points 1x, y2, where each x is an original sample value and y is the corresponding z score. Step 5: Examine the normal quantile plot and use the criteria given in Part 1. Conclude that the population has a normal distribution if the pattern of the points is reasonably close to a straight line and the points do not show some systematic pattern that is not a straight-line pattern. Dallas Commute Times EXAMPLE 1 Data set 31 “Commute Times” in Appendix B includes commute times (minutes) obtained from Dallas, Texas. Let’s consider this sample of the first five commute times: 20, 16, 25, 10, 30. With only five sample values, a histogram will not be very helpful here. Instead, construct a normal quantile plot for these five values and determine whether they appear to be from a population that is normally distributed. SOLUTION The following steps correspond to those listed in the procedure above for constructing a normal quantile plot. Step 1: First, sort the data by arranging them in order. We get 10, 16, 20, 25, 30. Step 2: With a sample of size n = 5, each value represents a proportion of 1>5 of the sample, so we proceed to identify the cumulative areas to the left of the corresponding sample values. The cumulative left areas, which are expressed in general as 1 2n , 3 2n , 5 2n, and so on, become these specific areas for this example with n = 5: 1 10 , 3 10 , 5 10 , 7 10, 9 10. These cumulative left areas expressed in decimal form are 0.1, 0.3, 0.5, 0.7, and 0.9. Step 3: We now use technology (or Table A-2) with the cumulative left areas of 0.1000, 0.3000, 0.5000, 0.7000, and 0.9000 to find these corresponding z scores: -1.28, -0.52, 0, 0.52, and 1.28. (For example, the z score of -1.28 has an area of 0.1000 to its left.)
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