Goodness-of-fit test. A statistical 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 m = $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.

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

b. Find the tabulated x2 value when a = .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. 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 $1,400 and $1,600 .136 136 143 Above $1,600 .023 23 13