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 χ2 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 deviations 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 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.

a. Compute the χ2 statistic on the basis of the observed and expected frequencies.
b. Find the tabulated χ2 value when α = .05 and there are five degrees of freedom. (There are k - 1 = 5 df associated with this χ2 statistic.)
c. On the basis of the χ2 statistic and the tabulated χ2 value, is there evidence that the salary distribution is nonnormal?
d. Find the approximate observed significance level for the test in part c.

Transcribed Image Text:

Expected Probability Frequency Frequency 23 136 341 341 136 23 Observed Interval .023 .136 .341 341 .136 .023 Less than $800 $1,000 < $1,200 26 146 < $1,000 361 $1,200 < $1,400 $1,400 < $1,600 311 143 13 or above