Lab Activity 9 – Comparing Two Groups | Complete Solution

Lab Activity 9 – Comparing Two Groups (53 points total)

Data set: College Scorecard Mideast

1.    (4 pts) Let p1 = population proportion of NY Mideast doctoral institutions that are private and p2 = population proportion of non-NY institutions that are private.  Test the claim that the proportion of Mideast doctoral schools that are private is different for NY and non-NY schools.  In other words, test H0: p1 – p2 = 0 versus Ha: p1 – p2 ≠ 0.  Perform the test using software.  Paste the output below.  Make sure to select the correct alternative hypothesis and test method (pooled estimate)!

a.    (2 pts) From the output, what is the test statistic, z, and the corresponding p-value?

b.    (4 pts)  Based on the p-value, do the data support the claim that the proportion of private schools differs in NY vs other Mideast states? Why or why not? Answer assuming α = .10.

c.    (2 pts) From the output, what is the pooled estimate of the proportion? (Note: if we reject H0, this is an estimate of the proportion of Mideast doctoral institutions that are private in both NY and other Mideast states.)

2.  Let’s determine if the average cost of school increased from 2010 to 2013 for Mideast doctoral institutions.

a.    (4 pts) Explain why we would use a paired samples test in this scenario.

b.    (3 pts) Using proper statistical notation, write down the null and alternative hypothesis for this test.  Define µd = µ1 – µ2, where µ1 is the mean cost of school in 2013 and µ2 is the mean cost of school in 2010.

Null hypothesis:   H0:
Alterative hypothesis:   Ha:

c.    (3 pts) Below are some descriptive statistics for ‘Cost_of_school_2013’ and ‘Cost_of_school_2010’ as obtained from Minitab Express.
 

Use this to calculate the test statistic for the test defined in part b (formula given below).  Assume that sd = 2226.8, and recall that  .  

d.    (1 pt) What are the degrees of freedom for this test statistic in part c? (See Section 9.5 of the online notes.)

e.    (4 pts) Use software to perform this test, and paste the output below.  Use it to double-check your results from above as well.

f.    (5 pts)  Decide between the null hypothesis and the alternative hypothesis based on the p-value and significance level, α = .01.  Then, write a sentence summarizing the real-world conclusion from your test.  Make sure your conclusion is specific and clear.

3.  Use the data to answer the question, “Does the mean completion rate (2013) differ for public and private schools?”  Assume 99% confidence (α = .01).

a.    (3 pts) Explain why we can NOT use a paired samples t-test in this scenario.

b.    (2 pts) Complete the correct notation for the null and alternative hypotheses for this test by filling in the two blanks below with either =, ≠, <, or >.  Note that µ1 is be the true mean completion rate (2013) for private schools, and µ2 is the true mean completion rate (2013) for public schools.

Null hypothesis:   H0: μ1 – μ2 ___ 0
Alterative hypothesis:   Ha: μ1 – μ2 ___ 0

c.    (2 pts) Should we perform a pooled or unpooled t-test? Explain how you came to this conclusion.  Note that sample standard deviation of completion rate in private schools is s1 = 0.15684 and the sample standard deviation of completion rate in public schools is s2 = 0.15488. (See Section 9.4 of the online notes.)

d.    (4 pts) Use software to perform the test from part b and paste the output below.  Make sure to select the correct alternative hypothesis and the correct test (pooled vs. unpooled, based on part c).

i.    (2 pts) From the output, what is the test statistic, t, and the corresponding p-value?

ii.    (5 pts)  Based on the p-value, what is your conclusion from the test?  Please state your decision to reject or fail to reject H0, and explain using the p-value.  Then, provide a sentence interpreting your findings in context of the problem.

iii.    (3 pts)  What is the confidence interval for the differences in the mean completion rate (2013) for the two groups?  Explain how this agrees with your conclusion for the test in part ii.