A poll of 1098 Americans showed that $47.2 \%$ of the respondents prefer to watch the news rather than read or listen to it. Use those results with a 0.10 significance level to test the claim that fewer than half of Americans prefer to watch the news rather than read or listen to it. Use the P-value method. Use the normal distribution as an approximation to the binomial distribution. Let $p$ denote the population proportion of all Americans who prefer to watch the news rather than read or listen to it. Identify the null and alternative hypotheses. \[ \begin{array}{ll} H_{0}: p & \nabla \\ H_{1}: p & \nabla \end{array} \] (Type integers or decimals. Do not round.)
Solution
Step 1 ：Let $p$ denote the population proportion of all Americans who prefer to watch the news rather than read or listen to it. We need to identify the null and alternative hypotheses.
Step 2 ：The null hypothesis is the statement that is being tested. In this case, the null hypothesis is that half of Americans prefer to watch the news rather than read or listen to it.
Step 3 ：The alternative hypothesis is the opposite of the null hypothesis. In this case, the alternative hypothesis is that fewer than half of Americans prefer to watch the news rather than read or listen to it.
Step 4 ：So, we have: \[\begin{array}{ll} H_{0}: p = 0.5 \\ H_{1}: p < 0.5 \end{array}\]
Step 5 ：Final Answer: \[\begin{array}{ll} H_{0}: p = \boxed{0.5} \\ H_{1}: p < \boxed{0.5} \end{array}\]
