Chapter 9: Q. 111 (page 555)

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?

Short Answer

Expert verified

There is sufficient evidence to conclude that more than $67 %$ of patrons borrow books in Owensboro, KY. The possible proportion of patrons that do borrow books from the library is $0.82$ .

Step by step solution

Given information

$x = 82, n = 100, p_{o} = 0.67, a = 0.01$

Explanation

State the hypothesis:

The null hypothesis states that most $67 %$ of the patrons borrow books in Owensboro, KY. In symbols:

$H_{o} : p \leq 0.67$

The alternative hypothesis states that more than $67 %$ of patrons borrow books in Owensboro, KY. In symbols:

$H_{a} : p > 0.67$

The sample proportion is:

$\overset{⏜}{P} = \frac{x}{n} = \frac{82}{100} = 0.82$

The test statistic is:

$z = \frac{\overset{⏜}{p} - p_{0}}{\sqrt{\frac{p_{0} (1 - p_{0})}{n}}} = \frac{0.82 - 0.67}{\sqrt{\frac{0.67 (1 - 0.67)}{100}}} = \frac{0.150}{0.0470} = 3.19$

For the right-tailed test, the p-value is:

$p - v a l u e = P (z > 3.19) = 0.0007$

Since $p - v a l u e = 0.0007 < 0.01$ , reject the null hypothesis.

There is sufficient evidence to conclude that $51.7 %$ of homes were heated by natural gas.

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