MGMT2263 Tutorial Sheet 2 Solutions

1) Since this is a two-tail test, it doesn’t matter which firm is group 1.

Ho: m1 = m2

Ha: m1 ¹ m2

Since we have two independent samples that are both at least 30 in size, we can use the Z test for two means.

Reject Ho if Z < -1.96 or > 1.96.

Z = (480 – 470)/sqrt(14²/30 + 10²/40) = 3.33.

Reject Ho.

Conclude there is a significant difference in the weekly wages of the two firms.

2) P-value = 2P(Z > 3.33) = 2(0.0004) = 0.0008. Since this is less than 1%, we have strong support for Ha. So, there is no level of significance between 1% and 10% we could have chosen in which the opposite conclusion would have been reached.

3) Lower limit = (480 – 470) – 1.96 sqrt(14²/30 + 10²/40) = 10 – 5.89 = 4.11

Upper limit = 10 + 5.89 = 15.89

4.11 < m(A) - m(B) < 15.89

Since the hypothesized difference of zero does not fall in the 95% confidence interval, we would reject Ho at a 5% level of significance.

If we use Megastat to solve questions 1 through 3, on the spreadsheet we would put the label, xbar, s and n for Firm A in column A and the similar information for Firm B in column B. From Hypothesis Tests, we would choose Compare Two Independent Groups. In the dialogue box, we would click the summary input radio button. The input range for group 1 would be A1:A4, that of group 2 would be B1:B4. The hypothesized difference is 0, and we choose not equal. We click the Z test radio button and check the box for the confidence interval. Here is the output:

Hypothesis Test: Independent Groups (z-test)
	Firm A	Firm B
	480	470	mean
	14	10	std. dev.
	30	40	n
		10.000	difference (Firm A - Firm B)
		3.006	standard error of difference
		0	hypothesized difference
		3.33	z
		.0009	p-value (two-tailed)
		4.1092	confidence interval 95.% lower
		15.8908	confidence interval 95.% upper
		5.891	margin of error

4) We need to solve –1.96 < (xbar1 – xbar2)/ sqrt(14²/30 + 10²/40) < 1.96 or xbar1 – xbar2 to fall between –5.89 and 5.89 (surprise, surprise, the margin of error from the confidence interval!). So, the largest difference between the two sample means in which we would not have rejected Ho is 5.89.

5) For this problem, we make the professionals group 1.

Ho: m1 - m2 £ 50

Ha: m1 - m2 > 50

Reject Ho if p-value < 1%. Do not reject Ho if p-value > 10%.

Z = (240 – 180 - 50)/sqrt(65²/400 + 45²/400) = 2.53.

p-value = P(Z > 2.53) = 0.5 – 0.4943 = 0.0057.

Since the p-value is less than 1%, reject Ho.

Conclude the average difference between professionals and the general public is more than 50 hours per month.

If we use Megastat, put the summary data for professionals in column A (label, xbar, s, n) and the summary data for public in column B. Follow the same procedure as above except the hypothesized difference is 50 and we choose right-tail. (You can also uncheck the confidence interval box.) Here is output:

Hypothesis Test: Independent Groups (z-test)
	Professional	Public
	240	180	mean
	65	45	std. dev.
	400	400	n
		60.000	difference (Professional - Public)
		3.953	standard error of difference
		50	hypothesized difference
		2.5298	z
		.0057	p-value (one-tailed, upper)