Chapter 9: Q 31. (page 565)

$18 %$ Members of the city council want to know if a majority of city residents supports a $1 %$ increase in the sales tax to fund road repairs. To investigate, they survey a random sample of $300$ city residents and use the results to test the following hypotheses:
$H_{0} : p = 0.50$
$H_{a} : p > 0.50$
where $p$ is the proportion of all city residents who support a $1 %$ increase in the sales tax to fund road repairs.
In the sample, $\hat{p} = 158 / 300 = 0.527$ , The resulting $P$ -value is $0.18$ . What is the correct interpretation of this $P$ -value?
a. Only $18 %$ of the city residents support the tax increase.
b. There is an $18 %$ chance that the majority of residents supports the tax increase.
c. Assuming that $50 %$ of residents support the tax increase, there is an $18 %$ probability that the sample proportion would be $0.527$ or greater by chance alone.
d. Assuming that more than $50 %$ of residents support the tax increase, there is an $18 %$ probability that the sample proportion would be $0.527$ or greater by chance alone.
e. Assuming that $50 %$ of residents support the tax increase, there is an $18 %$ chance that the null hypothesis is true by chance alone.

Expert verified

Option (c) is correct.

Step by step solution

It is given that $H_{0} : p = 0.50$

$H_{1} : p > 0.50$

$P = 0.18$

$\hat{p} = 158 / 300 = 0.527$

$P$ value refers to probability that proportion is more extreme as compared to sample proportion, type I error if null hypothesis is true.

If proportion of people residing in city who boost the tax is $0.50$ , null hypothesis is true.

The probability statement also needs inclusion of sample proportion.

Option (c) is correct.

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