A college student is interested in investigating the claim that students who graduate with a master's degree earn higher salaries, on average, than those who finish with a bachelor's degree. She surveys, at random, 33 recent graduates who completed their master's degrees and finds that their mean salary is [tex]$\$[/tex]32,200[tex]$ per year. The standard deviation of annual salaries for the population of recent graduates who have master's degrees is known to be $[/tex]\[tex]$2600$[/tex]. She also surveys, at random, 45 recent graduates who completed their bachelor's degrees and finds that their mean salary is [tex]$\$[/tex]31,400[tex]$ per year. The standard deviation of annual salaries for the population of recent graduates with only bachelor's degrees is known to be $[/tex]\[tex]$2100$[/tex]. Test the claim at the 0.01 level of significance. Let recent graduates with a master's degree be Population 1 and let recent graduates with a bachelor's degree be Population 2.

Step 1 of 3: State the null and alternative hypotheses for the test. Fill in the blank below.
[tex]$
\begin{array}{l}
H _0: \mu_1 = \mu_2 \\
H _a: \mu_1 \ \textgreater \ \mu_2
\end{array}
$[/tex]



Answer :

To correctly state the null and alternative hypotheses for this test, we need to clearly establish our assumptions about the population means for the two groups under consideration.

In this case:
- Population 1 consists of recent graduates with a master's degree.
- Population 2 consists of recent graduates with a bachelor's degree.

Null Hypothesis ([tex]\( H_0 \)[/tex]): The null hypothesis always states that there is no effect or no difference. Here, it means that the average salary of graduates with a master's degree is equal to the average salary of graduates with a bachelor's degree.
[tex]\[ H_0: \mu_1 = \mu_2 \][/tex]

Alternative Hypothesis ([tex]\( H_1 \)[/tex]): The alternative hypothesis states that there is a difference. In this context, the student is interested in investigating whether graduates with a master's degree earn higher salaries than those with a bachelor's degree. This implies a one-sided test where we are checking if the mean salary of master's degree graduates is greater than that of bachelor's degree graduates.
[tex]\[ H_1: \mu_1 > \mu_2 \][/tex]

Hence, the correct null and alternative hypotheses are:
[tex]\[ \begin{array}{l} H _0: \mu_1 = \mu_2 \\ H _1: \mu_1 > \mu_2 \end{array} \][/tex]

So the initial step for the hypothesis test is clearly defined.