Question

Question 4 In the past, 60% of all undergraduate students enrolled at a State University in NY earned their degrees within four years of matriculation. Administrators want to test the claim made by some faculty memebers that this proportion has fallen below the historical value of 60%. A random sample of 95 students from the class that matriculated in the fall of 2012 was recently selected to test this hypothesis. Administrators found that 44 of these 95 students graduated in the spring of 2016 (i.e., four academic years after matriculation). Use a significance level of 5%. a) Formulate the hypotheses. b) What is the value of the test statistic? c) What is the critical value of the test? d) Find the p-value of the test. e) What is your conclusion? Explain it in the context of the problem.

Answer 1

Solution :

Given that,

$p_{0}$ = 0.60

1 - $p_{0}$ = 0.40

n = 95

x = 44

Level of significance = $\alpha$ = 0.05

Point estimate = sample proportion = $\hat p$ = x / n = 0.46

This a left (One) tailed test.

a)

The null and alternative hypothesis is,

Ho: p = 0.60

Ha: p < 0.60

b)

Test statistics

z = ($\hat p$ - $p_{0}$ ) / $\sqrt{}$ $p_{0}$*(1-$p_{0}$) / n

= ( 0.46 - 0.60) / $\sqrt{}$ (0.60*0.40) / 95

= -2.723

c)

Critical value of  the significance level is α = 0.05, and the critical value for a two-tailed test is

$Z_{c}$ = -1.64

d)

P-value = P(Z<z)

=P(Z < -2.723)

= 0.0032

e)

The p-value is p = 0.0032, and since p = 0.0032 < 0.05, it is concluded that the null hypothesis is rejected.

Conclusion:

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim made by some faculty members that this proportion has fallen below the historical value of 60% at the α = 0.05 significance level.