A poll of 1012 teens aged 13 to 17 showed that $57 \%$ of them have made new friends online. Use a 005 significance level to test the claim that half of all teens have made new friends online. Use the P-value method. Use the normal distribution as an approximation to the binomial distribution. Let p denote the population proportion of all toens aged 13 to 17 who have made new friends online. Identify the null and alternative hypotheses. \[ \begin{array}{ll} H_{0}: P & \mathbf{Y} \\ H_{1}: P & \square \end{array} \] (Type integers or decimals. Do not round)
Solution
Step 1 ：Let p denote the population proportion of all teens aged 13 to 17 who have made new friends online. We need to identify the null and alternative hypotheses.
Step 2 ：The null hypothesis (H0) is the statement that we will test. It is usually the statement that there is no effect or difference. In this case, the null hypothesis is that half of all teens have made new friends online, so p = 0.5.
Step 3 ：The alternative hypothesis (H1) is the statement that we will accept if the data provide sufficient evidence that the null hypothesis is false. In this case, the alternative hypothesis is that the proportion of teens who have made new friends online is not 0.5.
Step 4 ：So, we have: H0: p = 0.5, H1: p ≠ 0.5
Step 5 ：Final Answer: \(\boxed{H_{0}: P = 0.5, H_{1}: P \neq 0.5}\)
