Question

The director of admissions at a large university says that 15% of high school juniors to whom she sends university literature eventually apply for admission. In a sample of 300 persons to whom materials were sent, 30 students applied for admission. In a two-tail test at the 0.05 level of significance, should we reject the director’s claim?

Answer 1

Let p be the population proportion of the students applied for admission among the students whom which the materials are sent. So the null and alternate hypotheses are given below.

$H_0:p=0.15 \\ H_1:p≠0.15$

The sample proportion is calculated below.

$\hat{p} = \frac{30}{300}=0.1$

The test statistic is calculated below.

$Z= \frac{\hat{p}-p_0}{ \sqrt{ \frac{p_0(1-p_0)}{n} } } \\ Z = \frac{0.10-0.15}{ \sqrt{ \frac{0.15(1-0.15)}{300} } } = -2.43$

The two tailed critical z value at 0.05 significance level is ±1.96 . Since the test statistic is less than the critical value -1.96, the null hypothesis will be rejected and hence the population proportion is different from 0. So the directors claim that "15% of high school juniors to whom she sends university literature eventually apply for admission" will be rejected.