Question

a consumer group claims that the mean minimum time it takes for a sedan to travel a quarter mile is greater than 14.7 seconds. a random sample of 22 sedans has a mean minimum time to travel a quarter mile of 15.5 seconds and a standard deviation of 2.11 seconds. at \\( \alpha = 0.10 \\) is there enough evidence to support the consumer groups claim? complete parts (a) through (d) below. assume the population is normally distributed.
(a) identify the claim and state \\( h _ { 0 } \\) and \\( h _ { a } \\).
\\( h _ { 0 } : \mu \leq 14.7 \\)
\\( h _ { a } : \mu > 14.7 \\)
(type integers or decimals. do not round.)
the claim is the alternative hypothesis.
(b) use technology to find the p - value. find the standardized test statistic, t.
\\( t = 1.78 \\)
(round to two decimal places as needed.)
obtain the p - value.
\\( p = \square \\)
(round to three decimal places as needed.)

Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom \(df=n - 1\), where \(n = 22\). So \(df=22-1=21\).

Step2: Use the t - distribution to find the P - value

Since this is a right - tailed test (because \(H_{a}:\mu>14.7\)), we use the t - distribution with \(df = 21\) and \(t=1.78\).
Using a t - table or technology (e.g., in Excel: =T.DIST.RT(1.78,21)), we calculate the P - value.

$$P=\text{T.DIST.RT}(1.78,21)$$

$$P\approx0.044$$

Answer:

\(P = 0.044\)