A student at a four-year college claims that average enrollment at four-year col

Question

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 5062 with a standard deviation of 4774. Of the 35 four-year colleges surveyed, the average enrollment was 5266 with a standard deviation of 8171. 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.)

State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom. Round your answer to two decimal places.)

(1) What is the test statistic? (Round your answer to two decimal places.)

(2) What is the p-value? (Round your answer to four decimal places.)

(3)Alpha: =

Explanation / Answer

1.

Let

u1 = mean of 4-year colleges
u2 = mean of 2-year colleges

Formulating the null and alternative hypotheses,              
              
Ho:   u1 - u2   <=   0  
Ha:   u1 - u2   >   0  
At level of significance =    0.05          
As we can see, this is a    right   tailed test.

We can use z distirbution here as both sample sizes, n1 and n2 are greater than 30.
      
Calculating the means of each group,              
              
X1 =    5266          
X2 =    5062          
              
Calculating the standard deviations of each group,              
              
s1 =    8171          
s2 =    4774          
              
Thus, the standard error of their difference is, by using sD = sqrt(s1^2/n1 + s2^2/n2):              
              
n1 = sample size of group 1 =    35          
n2 = sample size of group 2 =    35          
Also, sD =    1599.609926          
              
Thus, the z statistic will be              
              
z = [X1 - X2 - uD]/sD =    0.127531092 = 0.13   [ANSWER, TEST STATISTIC]

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2.      
              
Also, using p values, as this is right tailed, by table/technology,              
              
p =    0.449448229   [ANSWER, P VALUE]      
              
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3.

As given,

alpha = 0.05. [ANSWER]

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As P > 0.05, we fail to reject Ho.

There is no significant evidence that average enrollment at four-year colleges is higher than at two-year colleges in the United States. [CONCLUSION]

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