Question

Women athletes at the a certain university have a long-term graduation rate of 67%. Over the past several years, a random sample of 40 women athletes at the school showed that 22 eventually graduated. Does this indicate that the population proportion of women athletes who graduate from the university is now less than 67%? Use a 1% level of significance.

(a) State the null and alternate hypotheses. H0: p < 0.67; H1: p = 0.67 H0: p = 0.67; H1: p < 0.67 H0: p = 0.67; H1: p ≠ 0.67 H0: p = 0.67; H1: p > 0.67

(b) What sampling distribution will you use? The standard normal, since np < 5 and nq < 5. The standard normal, since np > 5 and nq > 5. The Student's t, since np > 5 and nq > 5. The Student's t, since np < 5 and nq < 5. What is the value of the sample test statistic? (Round your answer to two decimal places.)

(c) Find the P-value of the test statistic. (Round your answer to four decimal places.) and Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the true proportion of women athletes who graduate is less than 0.67. There is insufficient evidence at the 0.01 level to conclude that the true proportion of women athletes who graduate is less than 0.67.

Answer 1

Solution(a)
Given in the question
population proportion p = 0.67
Null hypothesis H0: p = 0.67
The claim is the population proportion of women athletes who graduates from the university now less than 67%
So alternate hypothesis Ha: p <0.67
So its correct answer is H0: p = 0.67 and Ha: p < 0.67
Solution(b)
Number of sample = 40
np = 40*0.67 = 26.8 > 5
nq = 40*0.33 = 13.2 > 5
So we will use standard normal since np>5 and nq>5
Solution(c)
sample proportionp^=X/N = 22/40 = 0.55
test statistic value can be calculated as
test stat = (p^ - p)/sqrt(p*(1-p)/n)) = (0.55-0.67)/sqrt(0.67*(1-0.67)/40) = -0.12/0.0743 = -1.61
From Z table we found P-value as this is left tailed test so P-value = 0.0533

Solution(d)
Based on the result, at alpha = 0.01, we are failed to reject the null hypothesis as thepvalue is greater than alpha value, and data are not statistically significant.
Solution(e)
So there is insufficient evidence at the 0.01 level to conclude that the true proportion of women athletes who graduate is less than 0.67.