Question

a college student is interested in investigating the claim that students who graduate with a masters degree earn higher salaries, on average, than those who finish with a bachelors degree. she surveys, at random, 34 recent graduates who completed their masters degrees, and finds that their mean salary is $39,800 per year. the standard deviation of annual salaries for the population of recent graduates who have masters degrees is known to be $900. she also surveys, at random, 31 recent graduates who completed their bachelors degrees, and finds that their mean salary is $39,000 per year. the standard deviation of annual salaries for the population of recent graduates with only bachelors degrees is known to be $1600. test the claim at the 0.05 level of significance. let recent graduates with a masters degree be population 1 and let recent graduates with a bachelors degree be population 2. step 3 of 3: draw a conclusion and interpret the decision. answer we reject the null hypothesis and conclude that there is insufficient evidence at a 0.05 level of significance to support the students claim that graduates with a masters degree earn higher salaries than those who finish with a bachelors degree. we fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.05 level of significance to support the students claim that graduates with a masters degree earn higher salaries than those who finish with a bachelors degree. we fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.05 level of significance to support the students claim that graduates with a masters degree earn higher salaries than those who finish with a bachelors degree. we reject the null hypothesis and conclude that there is sufficient evidence at a 0.05 level of significance to support the students claim that graduates with a masters degree earn higher salaries than those who finish with a bachelors degree.

Explanation:

Step1: Set up hypotheses

The null hypothesis $H_0:\mu_1\leq\mu_2$ and the alternative hypothesis $H_1:\mu_1 > \mu_2$, where $\mu_1$ is the population - mean salary of master's degree graduates and $\mu_2$ is the population - mean salary of bachelor's degree graduates.

Step2: Calculate the test - statistic

The formula for the two - sample z - test statistic when population standard deviations $\sigma_1$ and $\sigma_2$ are known is $z=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}}$. Here, $\bar{x}_1 = 39800$, $\bar{x}_2=39000$, $\sigma_1 = 900$, $\sigma_2 = 1600$, $n_1 = 34$, $n_2=31$, and under $H_0$, $\mu_1-\mu_2 = 0$. So, $z=\frac{(39800 - 39000)-0}{\sqrt{\frac{900^{2}}{34}+\frac{1600^{2}}{31}}}\approx\frac{800}{\sqrt{\frac{810000}{34}+\frac{2560000}{31}}}\approx\frac{800}{\sqrt{23823.53+82580.65}}\approx\frac{800}{\sqrt{106404.18}}\approx\frac{800}{326.19}\approx2.45$.

Step3: Determine the critical value and make a decision

For a one - tailed test with $\alpha = 0.05$, the critical value $z_{\alpha}=1.645$. Since $z = 2.45>1.645$, we reject the null hypothesis.

Answer:

We reject the null hypothesis and conclude that there is sufficient evidence at a 0.05 level of significance to support the student's claim that graduates with a master's degree earn higher salaries than those who finish with a bachelor's degree.