Question

For Americans using library services, the American Library Association claims that at most 67% of patrons borrow books. The library director in Owensboro, Kentucky feels this is not true, so she asked a local college statistic class to conduct a survey. The class randomly selected 100 patrons and found that 82 borrowed books. Did the class demonstrate that the percentage was higher in Owensboro, KY? Use α = 0.01 level of significance. What is the possible proportion of patrons that do borrow books from the Owensboro Library?

Answer 1

Answer:

The p-value of the test is 0.0007 < 0.01, which means that the class demonstrates that the percentage was higher in Owensboro, KY.

The possible proportion of patrons that do borrow books from the Owensboro Library is 0.82.

Step-by-step explanation:

For Americans using library services, the American Library Association claims that at most 67% of patrons borrow books. Test if the proportion is higher in Owensboro, KY.

At the null hypothesis, we test if the proportion is of at most 0.67, that is:

$H_0: p \leq 0.67$

At the alternative hypothesis, we test if the proportion is of more than 0.67, that is:

$H_1: p > 0.67$

The test statistic is:

$z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}$

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.

0.67 is tested at the null hypothesis:

This means that $\mu = 0.67, \sigma = \sqrt{0.67*0.33}$

The class randomly selected 100 patrons and found that 82 borrowed books.

This means that $n = 100, X = \frac{82}{100} = 0.82$

Value of the test statistic:

$z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}$

$z = \frac{0.82 - 0.67}{\frac{\sqrt{0.67*0.33}}{\sqrt{100}}}$

$z = 3.19$

P-value of the test and decision:

The p-value of the test is the probability of a finding a sample proportion of 0.82 or above, which is 1 subtracted by the p-value of z = 3.19.

Looking at the z-table, z = 3.19 has a p-value of 0.9993.

1 - 0.9993 = 0.0007

The p-value of the test is 0.0007 < 0.01, which means that the class demonstrates that the percentage was higher in Owensboro, KY.

What is the possible proportion of patrons that do borrow books from the Owensboro Library?

The sample proportion of 0.82.