Problem
The 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 142 out of a random sample of 220 residents in the community said they recycle, is there enough evidence to support the policy maker's claim at the 0.10 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 $H_{1}$. \[ \begin{array}{l} H_{0}: p=0.70 \\ H_{1}: p<0.70 \end{array} \] (b) Determine the type of test statistic to use.
Solution
Step 1 ：The null hypothesis $H_{0}$ is: $H_{0}: p=0.70$. This represents the claim that the proportion of residents who recycle is still 70%.
Step 2 ：The alternative hypothesis $H_{1}$ is: $H_{1}: p<0.70$. This represents the alternative claim that the proportion of residents who recycle is less than 70%.
Step 3 ：The type of test statistic to use in this case is a Z-test. This is because we are dealing with proportions and we have a large sample size (n > 30). The Z-test is used when the data is normally distributed, the standard deviation is known, and the sample size is large.
