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 in the next column. 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 s = $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.

Interval Probability Expected Frequency Observed Frequency Less than $800 .025 25 28 $800 6 $1,000 .150 150 155 $1,000 6 $1,200 .325 325 315 $1,200 6 $1,400 .325 325 312 $1,400 6 $1,600 .150 150 148 $1,600 or above .025 25 42

a. Compute the x2 statistic on the basis of the observed and expected frequencies.

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

c. On the basis of the x2 statistic and the tabulated x2 value, is there evidence that the salary distribution is non-normal?

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