Chapter 10.2, Problem 14E

Salaries of Recent College Graduates. The Tippie College of Business obtained the following results on the salaries of a recent graduating class:

1. a. Formulate hypothesis so that, if the null hypothesis is rejected, we can conclude that salaries for Finance majors are significantly lower than the salaries of Business Analytics majors. Use α = .05.
2. b. What is the value of the test statistic?
3. c. What is the p-value?
4. d. What is your conclusion?

To determine

Identify the null and alternative hypothesis.

Answer to Problem 14E

Null hypothesis:$H_0:{\mu }_1-{\mu }_2\ge 0$

Alternative hypothesis:$H_a:{\mu }_1-{\mu }_2<0$

Explanation of Solution

Calculation:

The results from a study of 110 salaries of Finance Majors and 30 salaries of Business Analytics Majors are as follows:

Finance Majors	Business Analytics Majors
$n_1=110$	$n_2=30$
${\overline{x}}_1=\$48,537$	${\overline{x}}_2=\$55,317$
$s_1=\$18,000$	$s_2=\$10,000$

$\alpha =0.05$

State the hypothesis:

The test hypotheses are to be obtained.

Null hypothesis:

$H_0:{\mu }_1-{\mu }_2\ge 0$

That is, the salaries of Finance majors are greater than or equal to the salaries of Business Analytics majors.

Alternative hypothesis:

$H_a:{\mu }_1-{\mu }_2<0$

That is, the salaries of Finance majors are less than the salaries of Business Analytics majors.

To determine

Find the value of test statistic.

Answer to Problem 14E

The value of the test statistic is –2.71.

Explanation of Solution

Calculation:

Test statistic:

The test statistic for hypothesis test of ${\mu }_1-{\mu }_2$ when ${\sigma }_1$ and ${\sigma }_2$ are unknown is as follows:

$t=\frac{\left({\overline{x}}_1-{\overline{x}}_2\right)-D_0}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$

Substitute ${\overline{x}}_1$ as 48,537, ${\overline{x}}_2$ as 55,317, $s_1$ as 18,000, $s_2$ as 10,000, $n_1$ as 110, and $n_2$ as 30 in the above formula given as follows:

$\begin{array}{c}t=\frac{\left(48,537-55,317\right)-0}{\sqrt{\frac{18,{000}^2}{110}+\frac{10,{000}^2}{30}}}\\ =\frac{-6,780}{\sqrt{2,945,454.5+3,333,333.3}}\\ =\frac{-6,780}{2,505.7}\\ =-2.71\end{array}$

Thus, the test statistic is –2.71.

To determine

Find the p-value.

Answer to Problem 14E

The p-value is 0.004.

Explanation of Solution

Calculation:

For t distribution with two independent random sample, the degrees of freedom is as follows:

$df=\frac{{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)}^2}{\frac{1}{n_1-1}{\left(\frac{s_1^2}{n_1}\right)}^2+\frac{1}{n_2-1}{\left(\frac{s_2^2}{n_2}\right)}^2}$

Substitute $s_1$ as 18,000, $s_2$ as 10,000, $n_1$ as 110, and $n_2$ as 30 in the above formula as follows:

$\begin{array}{c}df=\frac{{\left(\frac{18,{000}^2}{110}+\frac{10,{000}^2}{30}\right)}^2}{\frac{1}{110-1}{\left(\frac{18,{000}^2}{110}\right)}^2+\frac{1}{30-1}{\left(\frac{10,{000}^2}{30}\right)}^2}\\ =\frac{{\left(2,945,454.5+3,333,333.3\right)}^2}{\frac{1}{109}{\left(2,945,454.5\right)}^2+\frac{1}{29}{\left(3,333,333.3\right)}^2}\\ =\frac{\text{39,423,176,237,428}\text{.84}}{\text{79,593,598,271}\text{.3+383,141,754,789}\text{.3}}\\ =\frac{\text{39,423,176,237,428}\text{.84}}{\text{462,735,356,060}\text{.6}}\end{array}$

$\begin{array}{c}=85.2\\ \approx 85\end{array}$

Thus, the degrees of freedom for the t distribution is 85.

Software Procedure:

Step-by-step procedure to obtain the probability value using Excel:

• Open an EXCEL sheet and select the cell A1.
• Enter the formula =T.DIST(-2.71,85,TRUE) in the cell A1.
• Press Enter.

Output using the EXCEL software is given below:

From the output, the p-value is 0.004.

To determine

Obtain the conclusion.

Answer to Problem 14E

There is sufficient evidence to conclude that the salaries of Finance majors are less than the salaries of Business Analytics majors.

Explanation of Solution

Rejection rule:

If $p\text{-value}\le \alpha$, reject the null hypothesis.

If $p\text{-value}>\alpha$, do not reject the null hypothesis.

Conclusion:

From part (c), the p-value is 0.004.

Here, the p-value is less than the level of significance.

That is, $p\text{-value}\left(=0.004\right)<\alpha \left(=0.05\right)$.

By the rejection rule, the null hypothesis is rejected.

Therefore, there is sufficient evidence to conclude that the salaries of Finance majors are less than the salaries of Business Analytics majors.