Question

a data set about speed dating includes \like\ ratings of male dates made by the female dates. the summary statistics are ( n = 199, overline{x}=7.84, s = 1.97 ). use a 0.01 significance level to test the claim that the population mean of such ratings is less than 8.00. assume that a simple random sample has been selected. identify the null and alternative hypotheses, test statistic, p - value, and state the final conclusion that addresses the original claim.
what are the null and alternative hypotheses?
a. ( h_0:mu = 8.00 )
( h_1:mu<8.00 )
b. ( h_0:mu = 8.00 )
( h_1:mu>8.00 )
c. ( h_0:mu = 8.00 )
( h_1:mu
eq8.00 )
d. ( h_0:mu<8.00 )
( h_1:mu>8.00 )
determine the test statistic.
(round to two decimal places as needed.)
determine the p - value.
(round to three decimal places as needed.)
state the final conclusion that addresses the original claim.
( h_0 ). there is evidence to conclude that the mean of the population of ratings is 8.00.

Explanation:

Step1: Identify null and alternative hypotheses

The claim is that the population mean \(\mu\) is less than \(8.00\). The null hypothesis \(H_{0}\) is a statement of equality, so \(H_{0}:\mu = 8.00\). The alternative hypothesis \(H_{1}\) is the claim we are testing, so \(H_{1}:\mu<8.00\) (Option A).

Step2: Calculate the test - statistic

The formula for the \(t\) - test statistic (since the population standard deviation \(\sigma\) is unknown) is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\).
Given \(n = 199\), \(\bar{x}=7.84\), \(s = 1.07\), and \(\mu = 8.00\).

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Step3: Calculate the P - value

For a one - tailed \(t\) - test with \(n-1=199 - 1=198\) degrees of freedom and \(t=-2.11\). Using a \(t\) - distribution table or technology (e.g., a TI - 84 Plus: tcdf(-1000,-2.11,198)), the P - value is approximately \(0.018\).

Step4: Make a decision

The significance level \(\alpha = 0.01\). Since \(P - value=0.018>0.01=\alpha\), we fail to reject \(H_{0}\).

Answer:

• Null and alternative hypotheses: A. \(H_{0}:\mu = 8.00\), \(H_{1}:\mu<8.00\)
• Test statistic: \(-2.11\)
• P - value: \(0.018\)
• Final conclusion: Fail to reject \(H_{0}\). There is not sufficient evidence to conclude that the mean of the population of ratings is less than \(8.00\).