A data set about speed dating includes "like" ratings of male dates made by the female dates. The summary statistics are $n=185, \bar{x}=6.59, s=2.05$. Use a 0.05 significance level to test the claim that the population mean of such ratings is less than 7.00. Assume that a simple random sample has been selected. Identify the null and alternative hypotheses, test statistic, Pvalue, and state the final conclusion that addresses the original claim.
Determine the P-value.
(Round to three decimal places as needed.)
Answer
Final conclusion: There is sufficient evidence to support the claim that the population mean of 'like' ratings of male dates made by the female dates is less than 7.00. The test statistic is approximately \(\boxed{-2.72}\) and the P-value is \(\boxed{0.004}\).
Step 1 ：Identify the null and alternative hypotheses. The null hypothesis (H0) is that the population mean is 7.00, and the alternative hypothesis (H1) is that the population mean is less than 7.00.
Step 2 ：Calculate the test statistic using the formula: \[t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\] where \(\bar{x}\) is the sample mean, \(\mu_0\) is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size. The given values are n = 185, \(\bar{x}\) = 6.59, s = 2.05, \(\mu_0\) = 7.0. The calculated test statistic is approximately \(-2.72\).
Step 3 ：Determine the P-value. The P-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. The calculated P-value is \(0.004\), which is less than the significance level of 0.05.
Step 4 ：Since the P-value is less than the significance level, we reject the null hypothesis that the population mean is 7.00.
Step 5 ：Final conclusion: There is sufficient evidence to support the claim that the population mean of 'like' ratings of male dates made by the female dates is less than 7.00. The test statistic is approximately \(\boxed{-2.72}\) and the P-value is \(\boxed{0.004}\).
