Question

10. The National Collegiate Athletic Association (NCAA) requires colleges to report the graduation rates of their athletes. Here are data from a Big Ten university's report. For parts a and b, state your hypotheses, the test statistic and p-value, and then write a conclusion in terms of the problem.

(a) Ninety-five of the 147 athletes admitted in 1989-1991 graduated within six years. Did the percent of athletes who graduated within six years differ significantly from the all-university percentage, which was 70%?

(b) The graduation rates were 37 of 45 female athletes and 58 of 102 male athletes. Is there evidence that a smaller proportion of male athletes than of female athletes graduated within six years?

Answer 1

a)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.7
Alternative Hypothesis, Ha: p ≠ 0.7

Rejection Region
This is two tailed test, for α = 0.05
Critical value of z are -1.96 and 1.96.
Hence reject H0 if z < -1.96 or z > 1.96

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.6463 - 0.7)/sqrt(0.7*(1-0.7)/147)
z = -1.421

P-value Approach
P-value = 0.1553
As P-value >= 0.05, fail to reject null hypothesis.

b)

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 < p2

p1cap = X1/N1 = 58/102 = 0.5686
p1cap = X2/N2 = 37/45 = 0.8222
pcap = (X1 + X2)/(N1 + N2) = (58+37)/(102+45) = 0.6463

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.5686-0.8222)/sqrt(0.6463*(1-0.6463)*(1/102 + 1/45))
z = -2.96

P-value Approach
P-value = 0.0015
As P-value < 0.05, reject the null hypothesis.