Question

multiple choice 4 points
state your conclusion to the hypothesis test.
the mayor of a certain city is tasked with cutting over 5 million dollars out of next years budget and is considering cuts to spending on
public transportation. the mayor decides to conduct a hypothesis test and will only cut spending on public transportation if fewer than 15%
of residents use the service. in a random sample of 300 residents, 36 reported that they did use public transportation.
perform the appropriate hypothesis test at a 1% significance level.
reject the null hypothesis. there is sufficient evidence to conclude that fewer than 15% of residents use public transportation.
reject the null hypothesis. there is sufficient evidence to conclude that more than 15% of residents use public transportation.
do not reject the null hypothesis. there is insufficient evidence to conclude that more than 15% of residents use public transportation.
do not reject the null hypothesis. there is insufficient evidence to conclude that fewer than 15% of residents use public transportation.

Explanation:

Step1: Calculate the sample proportion

The sample proportion $\hat{p}=\frac{36}{300} = 0.12$

Step2: Set up the hypotheses

The null hypothesis $H_0:p = 0.15$ and the alternative hypothesis $H_1:p<0.15$ (since the mayor is checking if fewer than 15% use public transportation)

Step3: Calculate the test - statistic

The formula for the test - statistic in a one - sample proportion test is $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$
Substitute $\hat{p}=0.12$, $p = 0.15$, and $n = 300$
$z=\frac{0.12-0.15}{\sqrt{\frac{0.15\times(1 - 0.15)}{300}}}=\frac{- 0.03}{\sqrt{\frac{0.15\times0.85}{300}}}$
First, calculate the denominator: $\sqrt{\frac{0.15\times0.85}{300}}=\sqrt{\frac{0.1275}{300}}\approx\sqrt{0.000425}\approx0.0206$
Then, $z=\frac{-0.03}{0.0206}\approx - 1.46$

Step4: Find the critical value

For a one - tailed test with $\alpha = 0.01$, the critical value $z_{\alpha}$ is $z_{0.01}=- 2.33$ (from the standard normal distribution table)

Step5: Make a decision

Since $z=-1.46>-2.33$ (the test - statistic is not in the rejection region), we do not reject the null hypothesis.

Answer:

Do not reject the null hypothesis. There is insufficient evidence to conclude that fewer than 15% of residents use public transportation.