Question

consider a sample of 53 football games, where 28 of them were won by the home team. use a 0.10 significance level to test the claim that the probability that the home team wins is greater than one - half. identify the null and alternative hypotheses for this test. choose the correct answer below. a. $h_0:p = 0.5$, $h_1:p>0.5$ b. $h_0:p>0.5$, $h_1:p = 0.5$ c. $h_0:p = 0.5$, $h_1:p<0.5$ d. $h_0:p = 0.5$, $h_1:p
eq0.5$ identify the test statistic for this hypothesis test. the test statistic for this hypothesis test is 41 (round to two decimal places as needed.) identify the p - value for this hypothesis test. the p - value for this hypothesis test is 341 (round to three decimal places as needed.) identify the conclusion for this hypothesis test. a. reject $h_0$. there is not sufficient evidence to support the claim that the probability of the home team winning is greater than one - half. b. fail to reject $h_0$. there is sufficient evidence to support the claim that the probability of the home team winning is greater than one - half. c. reject $h_0$. there is sufficient evidence to support the claim that the probability of the home team winning is greater than one - half. d. fail to reject $h_0$. there is not sufficient evidence to support the claim that the probability of the home team winning is greater than one - half.

Explanation:

Step1: Hypothesis Testing Basics

In hypothesis testing, the null hypothesis \(H_0\) is a statement of no effect or equality. The alternative hypothesis \(H_1\) is what we are trying to find evidence for. Here, the claim is that the probability \(p\) of the home - team winning is greater than \(0.5\). So \(H_0:p = 0.5\) (no difference from one - half) and \(H_1:p>0.5\) (the claim we want to test).

Step2: Test Statistic and P - value Interpretation

The test statistic is a value calculated from the sample data. The P - value is the probability of getting a test statistic as extreme or more extreme than the one observed, assuming \(H_0\) is true. A large P - value (in this case, if \(P>0.10\) since the significance level \(\alpha = 0.10\)) means we fail to reject \(H_0\).
If \(P\leq\alpha\), we reject \(H_0\). If \(P>\alpha\), we fail to reject \(H_0\). Given a P - value (let's assume the correct P - value calculation, if we use the formula \(z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\), where \(\hat{p}=\frac{28}{53}\approx0.528\), \(p_0 = 0.5\), \(n = 53\), \(z=\frac{0.528 - 0.5}{\sqrt{\frac{0.5\times(1 - 0.5)}{53}}}\approx0.41\), and the P - value for \(z = 0.41\) (right - tailed test) is \(P=1-\Phi(0.41)\approx0.341\))
Since \(P - value=0.341>0.10=\alpha\)

Answer:

D. Fail to reject \(H_0\). There is not sufficient evidence to support the claim that the probability of the home team winning is greater than one - half.