Question

Fifteen hundred students attend Biship Ryan High School. 120 students were asked "Do you think that Library hours should be extended?" 40 students agreed that hours should be extended, 10 said they were against extended hours, and 70 students had no opinion. The school newspaper reported: "80% of students who had an opinion agree that school library hours should be extended". Is this a valid inference? Justify your reasoning.

Answer 1

The inference is valid.

Explanation:

In this case we need to test whether, 80% of students who had an opinion agree that school library hours should be extended.

A sample of 120 students were selected.

Of these 120, 50 students had an opinion and 70 did not.

Since we need to test for the students having an opinion, the sample size is, n = 50.

The sample proportion of students who had an opinion and agreed is:

`\hat p=(40)/(50)=0.80`

The hypothesis can be defined as follows:

H₀: The proportion of students who had an opinion agree that school library hours should be extended is 80%, i.e. p = 0.80.

Hₐ: The proportion of students who had an opinion agree that school library hours should be extended is different from 80%, i.e. p ≠ 0.80.

The z-test for single proportion will be used.

The test statistic is:

`z=\frac{\hat p-p}{\sqrt{(p(1-p))/(n)}}=\frac{0.80-0.80}{\sqrt{(0.80(1-0.80)/(50)}}=0`

The test statistic value is 0.

Compute the p-value of the test:

`p-value=2* P (Z < 0)\\=2* 0.50\\=1`

The p-value of the test is 1.

The p-value is very large, so the null hypothesis will not be rejected at any significance level.

Thus, it can be concluded that the proportion of students who had an opinion agree that school library hours should be extended is 80%.

Hence, the inference is valid.