294 CHAPTER 6 Normal Probability Distributions Key Concept The following chapters include important statistical methods requiring that sample data are from a population having a distribution that is approximately normal. In this section we present these steps for determining whether sample data satisfy the requirement of a normal distribution: 1. Construct a histogram and determine whether it is roughly bell-shaped. 2. Construct a normal quantile plot and use the criteria given later in this section. 6-5 Assessing Normality PART 1 Basic Concepts of Assessing Normality When trying to determine whether a collection of data has a distribution that is approximately normal, we can visually inspect a histogram to see if it is approximately bell-shaped (as discussed in Section 2-2), and we can also use a normal quantile plot (discussed briefly in Section 2-2). DEFINITION A normal quantile plot (or normal probability plot) is a graph of points 1x, y2 where each x value is from the original set of sample data, and each y value is the corresponding z score that is expected from the standard normal distribution. Procedure for Determining Whether It Is Reasonable to Assume That Sample Data Are from a Population Having a Normal Distribution 1. Histogram: Construct a histogram. If the histogram departs dramatically from a bell shape, conclude that the data do not have a normal distribution. 2. Normal quantile plot: If the histogram is basically symmetric, use technology to generate a normal quantile plot. Apply the following criteria to determine whether the distribution is normal. (These criteria can be used loosely for small samples, but they should be used more strictly for large samples.) Normal Distribution: The population distribution is normal 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. Not a Normal Distribution: The population distribution is not normal if either or both of these two conditions applies: ■ The points do not lie reasonably close to a straight-line pattern. ■ The points show some systematic pattern that is not a straight-line pattern. Advanced Methods: In addition to using histograms and normal quantile plots, there are other more advanced procedures for assessing normality, such as the Ryan-Joiner test (discussed briefly in Part 2 of this section). Other tests for normality include (insert drum roll here) the Shapiro-Wilk test, D’Agostino-Pearson test, chi-square goodness-of-fit test, Kolmogorov-Smirnov test, Lillefors corrected K-S test, Cramervon Mises test, Anderson-Darling test, the Jarque-Bera test, and the Anscombe-Glynn kurtosis test.
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