Question

Although union and non union wages tend to rise at the same rate in the long run, union wages usually advance faster during recessions and early in period of recovery, and non union wages tend to advance more rapidly later in the business cycle when the labor markets are tight. To examine this issue, an economist records the hourly wages (including employee benefits) of employees with two years experience for twelve chosen consumer products manufacturing firms. Six non union shops and six consumer manufacturing firms with union shops. The data (in dollars) are as follows:

Non union shops: 8.26, 8.17, 8.45, 9.09, 8.85, 8.31

Union shops: 7.92, 8.39, 8.64, 8.04, 8.24, 8.25

A) At 5% level of significance, does the data suggest that the union and non-union wages differ, on the average, for employees with two years experience in the consumer products manufacturing industry? Show all necessary work.

B) What type of error is possible and describe this error in terms of the problem?

C) Estimate the difference between the actual average wages of union and non-union employees with two years experience in the consumer products manufacturing industry using a 90% confidence interval. Does the data suggest that there is sufficient evidence of a difference in the actual average wage of union and non-union employees with 2 years experience in the consumer products manufacturing industry? Show all necessary work.

D) Interpret the interval estimation.

Answer 1

For Non union shops

= 8.5217, s1 = 0.3668, n1 = 6

For Union Shops

= 8.2467, s2 = 0.2548, n2 = 6

A) H₀:

    H₁:

The test statistic t = ()/sqrt(s1^2/n1 + s2^2/n2)

                             = (8.5217 - 8.2467)/sqrt((0.3668)^2/6 + (0.2548)^2/6)

                             = 1.508

df = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1))

    = ((0.3668)^2/6 + (0.2548)^2/6)^2/(((0.3668)^2/6)^2/5 + ((0.2548)^2/6)^2/5)

    = 9

At 0.05 significance level, the critical value is +/- t_0.025,9 = +/- 2.262

Since the test statistic value is not greater than the positive critical value(1.508 < 2.262), so we should not reject the null hypothesis.

No, the data does not suggest that the union and non-union wages differ, on the average for employees with two years experience in the consumer products manufacturing industry.

B) Type-II error is possible. The probability of not rejecting the null hypothesis that the average of union and non-union wages are equal when in fact the average of union and non-union wages are different.

C) At 90% confidence interval the critical value is t^* = 1.833

The 90% confidence interval is

() +/- t^* * sqrt(s1^2/n1 + s2^2/n2)

= (8.5217 - 8.2467) +/- 1.833 * sqrt((0.3668)^2/6 + (0.2548)^2/6)

= 0.275 +/- 0.3342

= -0.0592, 0.6092

The interval contains 0.

No, the data does not suggest that the union and non-union wages differ, on the average for employees with two years experience in the consumer products manufacturing industry.

D) We are 90% confident that the true difference in population means for the union and non-union wages for employees with two years experience in the consumer products manufacturing industry lies in the above confidence interval.