Question

a data set about speed dating includes like ratings of male dates made by the female dates. the summary statistics are ( n = 106 ), ( overline{x}=6.80 ), ( s = 2.27 ). 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, p - value, and state the final conclusion that addresses the original claim.
what are the null and alternative hypotheses?
a ( h_{0}:mu = 7.00 )
( h_{1}:mu<7.00 )
b ( h_{0}:mu = 7.00 )
( h_{1}:mu>7.00 )
c ( h_{0}:mu = 7.00 )
( h_{1}:mu
eq7.00 )
d ( h_{0}:mu<7.00 )
( h_{1}:mu = 7.00 )
determine the test statistic.
(round to two decimal places as needed)

Explanation:

Step1: State the null and alternative hypotheses

The null hypothesis \(H_{0}\) is a statement of equality. The claim is that the population mean is less than \(7.00\). So, \(H_{0}:\mu = 7.00\) (equality) and \(H_{1}:\mu<7.00\) (the claim we are testing).

Step2: Calculate the test - statistic

The formula for the \(t\) - test statistic (since the population standard deviation \(\sigma\) is unknown) is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\)

We are given \(\bar{x} = 6.80\), \(\mu = 7.00\), \(s = 2.22\), \(n = 106\)

$$
LATEXBLOCK0
$$

Step3: Find the \(P\) - value

For a one - tailed \(t\) - test with \(n - 1=106 - 1 = 105\) degrees of freedom and \(t=-0.93\). Using a \(t\) - distribution table or technology (e.g., a TI - 84 Plus: tcdf(-1E99,-0.93,105)), the \(P\) - value is approximately \(0.177\)

Step4: Make a conclusion

Since the \(P\) - value (\(0.177\))> significance level (\(\alpha = 0.05\)), we fail to reject the null hypothesis.

There is not sufficient evidence at the \(0.05\) significance level to support the claim that the population mean of such ratings is less than \(7.00\)

Answer:

A. \( H_{0}:\mu = 7.00\), \(H_{1}:\mu<7.00\)