[please show work] The dean of the College of Business at the University of La Verne has observed for several years and found that the probability

[please show work] The dean of the College of Business at the University of La Verne has observed for several years and found that the probability distribution of the salary of the alumni's first job after graduation is normal. The college collected information from 250 alumni and finds that the mean of their salary is $50k. Assuming a 95% confidence level, please do the following:

Suppose the standard deviation of the collected data from 250 alumni is $10k. Can we conclude that the true mean of alumni salary is different from $55k?

(a) What are the null and alternate hypotheses ?

(b) Decide on the test statistic and calculate the value of the test statistic (hint: write the equation and calculate the statistic

(c) Please make the conclusion 

Now, suppose that the standard deviation of the collected data from 250 alumni is $10k and the mean is $50k, please estimate the confidence interval of the population's average salary 

The dean wants to know whether there is salary difference between male and female alumni. He knows that the population standard deviation for male is $8 and for female is $10. The sampled data show that the average salary for 100 male is $56 and the average salary for 121 female is $52.

(a) what are the null & alternate hypothesis?

(b) Decide on the test statistic and calculate the value of the test statistic (hint: write the equation and calculate the statistic) 

(c) What's your decision regarding the hypothesis and interpret the result using test-score rejection region rule or p value rule