Question

$18\%$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.

Short answer

Option (c) is correct.

Step-by-step solution

Given Information

It is given that $H_0:p=0.50$

$H_1:p>0.50$

$P=0.18$

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

Explanation

$P$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.