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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 :

GinnyAnswer
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.