A student at a four-year college claims that average enrollment at four-year colleges is higher than at two-year colleges in the United States. Two surveys are conducted. Of the 35 two-year colleges surveyed, the average enrollment was 5069 with a standard deviation of 4773. Of the 35 four-year colleges surveyed, the average enrollment was 5216 with a standard deviation of 8141. Conduct a hypothesis test at the 5% level. NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)Required:a. State the distribution to use for the test. b. What is the test statistic?c. What is the p-value?

(a) The test statistics that would be used here Two-sample t-test statistics distribution.

(b) The value of t-test statistic is 0.092.

(c) P-value of the test statistics is more than 40%.

Step-by-step explanation:

We are given that of the 35 two-year colleges surveyed, the average enrollment was 5069 with a standard deviation of 4773.

Of the 35 four-year colleges surveyed, the average enrollment was 5216 with a standard deviation of 8141.

Let = average enrollment at four-year colleges in the United States.

= average enrollment at two-year colleges in the United States.

So, Null Hypothesis, :      {means that the average enrollment at four-year colleges is higher than at two-year colleges in the United States}

Alternate Hypothesis, :      {means that the average enrollment at four-year colleges is higher than at two-year colleges in the United States}

(a) The test statistics that would be used here Two-sample t-test statistics distribution because we don't know about population standard deviation;

                              T.S. =    ~

where, = average enrollment at four-year colleges = 5216

= average enrollment at two-year colleges = 5069

= sample standard deviation at four-year colleges = 8141

= sample standard deviation at two-year colleges = 4773

= sample of four-year colleges surveyed = 35

=  sample of two-year colleges surveyed = 35

Also,  =   = 6672.98

So, the test statistics  =    ~

                                     =  0.092

(b) The value of t-test statistic is 0.092.

(c) P-value of the test statistics is given by the following formula;

                 P-value = P( > 0.092) = More than 40% as this value is not reflected in the t-table.

x = 3 ± √(5+y)