Question

A university interested in tracking its honors program believes that the proportion of graduates with a GPA of 3.00 or below is less than 0.16. In a sample of 200 graduates, 24 students have a GPA of 3.00 or below. The value of the test statistic and its associated p-value at the 5% significance level are _________, respectively.

Answer 1

The value of the test statistic and its associated p-value at the 5% significance level are -1.54 and 0.9382, respectively.

Explanation:

A university interested in tracking its honors program believes that the proportion of graduates with a GPA of 3.00 or below is less than 0.16.

This means that the null hypothesis is:

`H_(0): p \geq 0.16`

Testing this hypothesis, means that the alternate hypothesis is:

`H_(a): p > 0.16`

The test statistic is:

`z = (X - \mu)/((\sigma)/(√(n)))`

In which X is the sample mean,
`\mu` is the value tested at the null hypothesis,
`\sigma` is the standard deviation and n is the size of the sample.

0.16 is tested at the null hypothesis:

This means that
`\mu = 0.16, \sigma = √(0.16*0.84)`

In a sample of 200 graduates, 24 students have a GPA of 3.00 or below.

This means that
`n = 200, X = (24)/(200) = 0.12`

Value of the test statistic:

`z = (X - \mu)/((\sigma)/(√(n)))`

`z = (0.12 - 0.16)/((√(0.16*0.84))/(√(200)))`

`z = -1.54`

Pvalue:

The pvalue is the probability of finding a sample mean above 0.12, which is 1 subtracted by the pvalue of z = -1.54.

Looking at the z-table, z = -1.54 has a pvalue of 0.0618

1 - 0.0618 = 0.9382

The value of the test statistic and its associated p-value at the 5% significance level are -1.54 and 0.9382, respectively.