Chapter 10, Problem 10.57ACA

Goodness-of-fit test. A statistics analysis is to be done on a set of data consisting of 1,000 monthly salaries. The analysis requires the assumption that the sample was drawn from a normal distribution. A preliminary test, called the x² goodness-of-fit test, can be used to help determine whether it is reasonable to assume that the sample is from a normal distribution. Suppose the mean and standard deviation of the 1.000 salaries are hypothesized to be $1.200 and $200, respectively. Using the standard normal table, we can approximate the probability of a salary being in the intervals listed in the table below. The third column represents the expected number of the 1.000 salaries to be found in each interval if the sample was drawn from a normal distribution with μ – $1,200 and Σ – $200. Suppose the last column contains the actual observed frequencies in the sample. Large differences between the observed and expected frequencies cast doubt on the normality assumption.

Table for Exercise 10.57

Interval	Probability	Expected Frequency	Observed Frequency
Less than $800	.023	23	26
Between $800 and $1,000	.136	136	146
Between $1,000 and $1,200	.341	341	361
Between $1,200 and $1.400	.341	341	311
Between $1,430 and $1,600	.136	136	143
Above $1,630	.023	23	13

a. Compute the x² statistic based on the observed and expected frequencies—just as you did n Section 10.2 α – .01.

b. Find the tabulated x² value when α = .05 and there are 5 degrees of freedom. (There are k: 1 = 5 df associated with this x² statistic.)

c. Based on the x² statistic and the tabulated x² value, is there evidence that the salary distribution is nonnormal?

d. Find an approximate observed significance level for the test in part c.