Table 7 in the Appendix is useful for obtaining percentiles for the chi-square distribution for a wide range of values of degrees of freedom and values of α. Alternatively, when a computer is available, a software program such as R can be used to obtain percentiles or to compute p-values for values of degrees of freedom and values of a not provided in Table 7. However, in those situations when a computer is unavailable or Table 7 does not have the desired percentile for a specified degrees of freedom, the following approximation can be used provided df > 40.

where x2a is the upper percentile of the chi-square distribution with df = v and za is the upper a percentile of a standard normal distribution.
a. For df = 20, compare the values obtained from the approximation for x2.05 and x2.95 to the values listed in Table 7.
b. For df = 60, compare the values obtained from the approximation for x2.05 and x2.95 to the values listed in Table 7.
c. For df = 90, compare the values obtained from the approximation for x2.05 and x2.95 to the values listed in Table 7.
d. For df = 240, compare the values obtained from the approximation for x2.05 and x2.95 to the values listed in Table 7.
e. Comment on the accuracy of the approximation for the percentiles obtained in parts (a)€“(d).

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