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, 49
 recent graduates who completed their master’s degrees, and finds that their mean salary is $31,700
 per year. The standard deviation of annual salaries for the population of recent graduates who have master’s degrees is known to be $1500
. She also surveys, at random, 41
 recent graduates who completed their bachelor’s degrees, and finds that their mean salary is $30,700
 per year. The standard deviation of annual salaries for the population of recent graduates with only bachelor’s degrees is known to be $2300
. Test the claim at the 0.05
 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 2 of 3 :  Compute the value of the test statistic. Round your answer to two decimal places.

We need to perform a hypothesis test for the difference between two population means, given that the population standard deviations are known. This suggests we should use a two-sample Z-test for the difference in means.

### Step 1: State the null and alternative hypotheses

- **Null hypothesis** ($H_0$): There is no difference in the mean salaries of master's and bachelor's degree graduates.
  \[
  H_0: \mu_1 = \mu_2 \quad \text{or equivalently} \quad \mu_1 - \mu_2 = 0
  \]
- **Alternative hypothesis** ($H_a$): The mean salary of master's degree graduates is higher than that of bachelor's degree graduates.
  \[
  H_a: \mu_1 > \mu_2 \quad \text{or equivalently} \quad \mu_1 - \mu_2 > 0
  \]

This is a right-tailed test.

### Step 2: Compute the test statistic

The test statistic for a two-sample Z-test is calculated using the formula:

\[
Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}
\]

Where:
- $\bar{x}_1 = 31,700$ (sample mean of master's degree graduates)
- $\bar{x}_2 = 30,700$ (sample mean of bachelor's degree graduates)
- $\sigma_1 = 1,500$ (population standard deviation of master's degree graduates)
- $\sigma_2 = 2,300$ (population standard deviation of bachelor's degree graduates)
- $n_1 = 49$ (sample size of master's degree graduates)
- $n_2 = 41$ (sample size of bachelor's degree graduates)
- $\mu_1 - \mu_2 = 0$ (under the null hypothesis)

Substitute the values into the formula:

\[
Z = \frac{(31,700 - 30,700) - 0}{\sqrt{\frac{1,500^2}{49} + \frac{2,300^2}{41}}}
\]

Now let's calculate this value.The value of the test statistic $ Z $ is approximately 2.39.

Would you like further details on this calculation, or have any additional questions?

Here are some related questions:
1. How do you interpret a Z-test result?
2. What is the next step after calculating the Z-test statistic?
3. How is the level of significance used in hypothesis testing?
4. What is the difference between a one-tailed and a two-tailed test?
5. How would the test change if the population standard deviations were unknown?

**Tip**: Always check if your test is one-tailed or two-tailed before interpreting the result.

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, 49 recent graduates who completed their master’s degrees, and finds that their mean salary is $31,700 per year. The standard deviation of annual salaries for the population of recent graduates who have master’s degrees is known to be $1500. She also surveys, at random, 41 recent graduates who completed their bachelor’s degrees, and finds that their mean salary is $30,700 per year. The standard deviation of annual salaries for the population of recent graduates with only bachelor’s degrees is known to be $2300. Test the claim at the 0.05 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 2 of 3: Compute the value of the test statistic. Round your answer to two decimal places.

Test the claim that master’s degree graduates earn higher salaries than bachelor’s degree graduates using a two-sample Z-test. This problem includes sample means, population standard deviations, and a step-by-step calculation of the test statistic with a significance level of 0.05.

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