Question

the national collegiate athletic association requires colleges to report the graduation rates of their athletes. at one large university, 78% of all students who entered in 2004 graduated in six years. one hundred thirty seven of the 190 students who entered with athletic scholarship graduated . consider these 190 as a sample of all athletes who will be admitted under present policies. is there evidence at the 5% level that the the percentage of athletes who graduate is less than 78%?

1. List the conditions for the test you plan to use, explain how these conditions are met, calculate and write down the test statistic, and find the P-value.

2. based on your P-value, conclude in context

Answer 1

(1) Conditions:

(a) The Sampling is a Simple random sampling

(b) Normality: n * p = 190*0.78 = 148.2 and n * (1-p) = 190*0.22 = 41.8 are both greater than 10.

(c) Each response in independent of the others.

(d) 10% Condition 190 students comes from a population of at least 1900 students who get admission with athletic scholarship.

(2) Hypothesis Test For a Single Proportion

Let = The sample proportion of athletes who got admission through athletic scholarship and graduated = 137/190 = 0.721

Let p = The population proportion of athletes who got admission through athletic scholarship and graduated = 78% = 0.78

1 - p = 0.22

= 0.05

The Hypothesis:

H0: p = 0.78: The population proportion of athletes who graduated after getting admission through an athletic scholarship is equal to 0.78.

Ha: p < 0.78: The population proportion of athletes who graduated after getting admission through an athletic scholarship is lesser than 0.78.

This is a 2 Tailed Test.

The Test Statistic:

Z observed = -1.96

The p Value: The p value (1 Tail) for Z = -1.96, is; p value = 0.025

The Critical Value: The critical value (1 tail) at = 0.05, Zcritical = -1.645

The Decision Rule:    

The Critical Value Method: If Zobserved is < -Zcritical, Then Reject H0.

The p value Method: If the P value is < , Then Reject H0

The Decision:   

The Critical Value Method: Since Z observed (-1.96) < -Zcritical (-1.645), We Reject H0

The p value Method: Since P value (0.025) is < (0.05), We Reject H0.

The Conclusion: There is sufficient evidence at the 95% significance level to conclude that the population proportion of of athletes who graduated after getting admission through an athletic scholarship is lesser than 0.78.

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