580 CHAPTER 11 Goodness-of-Fit and Contingency Tables Good fit with assumed distribution Not a good fit with assumed distribution Compare the observed O values to the corresponding expected E values. Os and Es are close Os and Es are far apart “If the P is low, the null must go.” Fail to reject H0 Reject H0 Small x2 value, large P-value Large x2 value, small P-value x2 here x2 here FIGURE 11-1 Relationships Among the X2 Test Statistic, P-Value, and Goodness-of-Fit The x2 test statistic is based on differences between the observed and expected values. If the observed and expected values are close, the x2 test statistic will be small and the P-value will be large. If the observed and expected frequencies are far apart, the x2 test statistic will be large and the P-value will be small. Figure 11-1 summarizes this relationship. The hypothesis tests of this section are always right-tailed, because the critical value and critical region are located at the extreme right of the distribution. If you are confused, just remember this mnemonic: “If the P is low, the null must go.” (If the P-value is small, reject the null hypothesis that the distribution is as claimed.) Measured or Reported? EXAMPLE 1 It is well known that when people report data, the results can be very different from those that are measured, especially when sensitive data such as weights or heights are involved. When weights of people are measured, the last digits of those weights tend to occur with about the same frequency. Data Set 4 “Measured and Reported” includes weights of subjects that were both reported and measured, and the data are from the National Center for Health Statistics. The last digits of weights (lb) of males are summarized in Table 11-2. Test the claim that the sample is from a population of weights in which the last digits do not occur with the same frequency. Based on the results, does it appear that the weights were reported or measured? Go Figure 3.141592653: The first 10 digits of p, which is a number with an infinite number of digits that have no pattern of repetition. TABLE 11-2 Last Digits of Weights of Males Last Digit 0 1 2 3 4 5 6 7 8 9 Frequency 1175 44 169 111 112 731 96 110 171 65
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