Question

a consumer group claims that the mean minimum time it takes for a sedan to travel a quarter mile is greater than 14.8 seconds. a random sample of 21 sedans has a mean minimum time to travel a quarter mile of 15.5 seconds and a standard deviation of 2.12 seconds. at \\( \alpha = 0.05 \\) 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 14.8 \\)
\\( h _ { a } : \mu > 14.8 \\)
(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 = 1.51 \\)
(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 \(n = 21\). So \(df=21-1 = 20\)

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

We have a one - tailed t - test (right - tailed since \(H_{a}:\mu>14.8\)) with \(t = 1.81\) and \(df = 20\).
Using a t - distribution table or technology (e.g., in R: 1 - pt(1.81, 20) or in Excel: =TDIST(1.81,20,1)), we calculate the P - value.
The formula for the P - value in a one - tailed t - test is \(P(T>t)\) where \(T\sim t(df)\)
\(P\approx0.042\)

Answer:

\(P = 0.042\)