Question

a data set about speed dating includes \like\ ratings of male dates made by the female dates. the summary statistics are ( n = 191 ), ( overline{x}=6.51 ), ( s = 1.96 ). use a 0.10 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
eq7.00 )
b. ( h_0:mu = 7.00 )
( h_1:mult7.00 )
c. ( h_0:mu = 7.00 )
( h_1:mugt7.00 )
d. ( h_0:mult7.00 )
( h_1:mugt7.00 )
determine the test statistic.
(round to two decimal places as needed.)

Explanation:

Step1: Calculate the test statistic

The formula for the test statistic \(t\) in a one - sample \(t\) - test is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\).
Here, \(\bar{x} = 6.51\), \(\mu = 7.00\), \(s = 1.96\), and \(n=191\).
Substitute the values into the formula:

$$
LATEXBLOCK0
$$

First, calculate \(\sqrt{191}\approx13.82\), then \(1.96/\sqrt{191}\approx1.96/13.82\approx0.142\)

$$ t=\frac{-0.49}{0.142}\approx - 3.45 $$

Answer:

The test statistic is approximately \(-3.45\)