Confidence Intervals and Hypothesis Testing
Hypothesis test for a population proportion using the critical value method
Haile
a proportion $p$ of residents in a community who recycle has traditionally been $70 \%$. A policy maker claims that the proportion is less than $70 \%$ now that one of the recycling centers has been relocated. If 152 out of a random sample of 240 residents in the community said they recycle, is there enough evidence to support the policy maker's claim at the 0.05 level of significance?
Perform a one-tailed test. Then complete the parts below.
Carry your intermediate computations to three or more decimal places. (If necessary, consult a list of formulas.)
(a) State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$.
\[
\begin{array}{l}
H_{0}: \square \\
H_{1}: \square
\end{array}
\]
(b) Determine the type of test statistic to use.
(Choose one) $\mathbf{v}$
(c) Find the value of the test statistic. (Round to three or more decimal places.) $\square$
(d) Find the critical value. (Round to three or more decimal places.)
(e) Is there enough evidence to support the policy maker's claim that the proportion of residents who recycle is less than $70 \%$ ?
\begin{tabular}{ccc}
\hline$\mu$ & $\sigma$ & $p$ \\
$\bar{x}$ & $s$ & $\hat{p}$ \\
$\square^{\square}$ & $\square_{\square}$ & $\frac{\square}{\square}$ \\
$\square=\square$ & $\square \leq \square$ & $\square \geq \square$ \\
$\square \neq \square$ & $\square< \square$ & $\square> \square$ \\
$x$ & & $S$
\end{tabular}
Answer
Final Answer: \(\boxed{H_{0}: p = 0.7}\), \(\boxed{H_{1}: p < 0.7}\), \(\boxed{\text{Test statistic (z-score)}: -2.254}\), \(\boxed{\text{Critical value}: -1.645}\), \(\boxed{\text{There is enough evidence to support the policy maker's claim that the proportion of residents who recycle is less than 70%}}\)
Step 1 ：State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$. The null hypothesis $H_{0}$: $p = 0.7$. The alternative hypothesis $H_{1}$: $p < 0.7$.
Step 2 ：Determine the type of test statistic to use. The type of test statistic to use is a z-score.
Step 3 ：Find the value of the test statistic. The value of the test statistic is -2.254.
Step 4 ：Find the critical value. The critical value is -1.645.
Step 5 ：Is there enough evidence to support the policy maker's claim that the proportion of residents who recycle is less than $70 \%$ ? Yes, there is enough evidence to support the policy maker's claim that the proportion of residents who recycle is less than 70%.
Step 6 ：Final Answer: \(\boxed{H_{0}: p = 0.7}\), \(\boxed{H_{1}: p < 0.7}\), \(\boxed{\text{Test statistic (z-score)}: -2.254}\), \(\boxed{\text{Critical value}: -1.645}\), \(\boxed{\text{There is enough evidence to support the policy maker's claim that the proportion of residents who recycle is less than 70%}}\)
