In the last 4 years a total of 55 Gossett High School studen

In the last 4 years a total of 55 Gossett High School students were admitted to nearby Chapel Hillside University. Of those students, 16 were offered athletic scholarships. Gossett High Schools guidance counselor looked at the composite ACT scores of the 55 students admitted to Chapel Hillside University (summary shown in the table), wondering if Chapel Hillside University might admit students with lower ACT scores if they were also athletes. Is there evidence that the mean ACT score of athletes admitted to Chapel Hillside University is significantly lower than the mean ACT score of non-athletes admitted to Chapel Hillside University?

Let _A denote the mean ACT score of all athletes admitted to Chapel Hillside University and _Na denote the mean ACT score of all non-athletes admitted to Chapel Hillside University.

Question 1. What are the appropriate null and alternative hypotheses for this problem?

H₀:_ANa<0, H_a:_ANa>0

H₀:_ANa=0, H_a:_ANa0    

H₀:_ANa=0, H_a:_ANa>0

H₀:_ANa = 0, H_a:_ANa<0

H₀:_ANa>0, H_a:_ANa<0

Question 2. Determine the value of the test statistic for this hypothesis test (do not assume equal variances) Use 3 decimal places in your answer.

Question 3. What is the rejection region if the significance level is = .05? Use 25 degrees of freedom. (right click on this link to open a t-table in a new window).

t < -1.316

t < -1.708    

t > 1.708

t > 2.060

t < -2.060

Question 4. What is the conclusion of this hypothesis test in terms of the problem?

Do not reject the null hypothesis; there is no significant difference between the mean ACT scores of athletes and non-athletes.

Even though the mean ACT score of the 16 athletes is less than the mean ACT score of the 39 non-athletes, the difference is not enough to be statistically meaningful.    

Conclude that there is a .95 chance that the mean ACT score of athletes is lower than the mean ACT score of non-athletes.

Reject the null hypothesis that there is no difference in the mean ACT scores and conclude that the mean ACT score of the athletes is lower than the mean ACT score of the non-athletes.

Conclude that there is a .05 chance that the mean ACT score of athletes is lower than the mean ACT score of non-athletes.

Solution

Sol)

Given Claim Is there evidence that the mean ACT score of athletes admitted to Chapel Hillside University is significantly lower than the mean ACT score of non-athletes admitted to Chapel Hillside University

1) Null hypothesis:H0:ANa = 0,

Alternative hyp: Ha:ANa<0 (left tail)

2) Test Statistics

t= -3.075

3) t critical value at 5% level at 25 df is -1.708

ie t < -1.708 or t > 1.708

4) t cal > t critical value Hence we reject the null hypothesis

Reject the null hypothesis that there is no difference in the mean ACT scores and conclude that the mean ACT score of the athletes is lower than the mean ACT score of the non-athletes.