Question

The accompanying table shows the annual​ salaries, in thousands of​ dollars, earned by a sample of individuals who graduated with MBAs in the years 2012 and 2013. Perform a hypothesis test using α equals= 0.05 to determine if the median salary for 2012 MBA graduates is lower than 2013 MBA graduates.

Salaries​ (in thousands of​ dollars) for 2012 MBA graduates

92.6	66.1	60.6	85.5	92.8	44.7	84.4
81.1	44.2	94.3	66.6	61.6	94.6	81.6

Salaries​ (in thousands of​ dollars) for 2013 MBA graduates

64.4	39.4	53.9	84.9	43.3	35.6	70.3	62.8
35.4	76.9	54.5	59.1	66.3	57.2	64.7

Identify the test statistic.

___ ​(Round to two decimal places as​ needed.)

Calculate the​ p-value.

​p-value equals=____ (Round to three decimal places as​ needed.)

Answer 1

The data is given by:

	Salaries​ (in thousands of​ dollars) for 2012 MBA graduates	Salaries​ (in thousands of​ dollars) for 2013 MBA graduates
	92.6	64.4
	66.1	39.4
	60.6	53.9
	85.5	84.9
	92.8	43.3
	44.7	35.6
	84.4	70.3
	81.1	62.8
	44.2	35.4
	94.3	76.9
	66.6	54.5
	61.6	59.1
	94.6	66.3
	81.6	57.2
		64.7
Count	14	15
Average	75.05	57.91333333
StDev	17.68531551	14.67567797

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: ​

Ha: ​

This corresponds to a left-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=27. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

Hence, it is found that the critical value for this left-tailed test is tc​=−1.703, for α=0.05 and df=27.

The rejection region for this left-tailed test is R={t:t<−1.703}.

(3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

The p-value is p=0.9958

(4) Decision about the null hypothesis

Since it is observed that t=2.847≥tc​=−1.703, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.9958, and since p=0.9958≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is less than μ2​, at the 0.05 significance level.

Graphically

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