Question

a consumer group claims that the mean minimum time it takes for a sedan to travel a quarter mile is greater than 14.6 seconds. a random sample of 22 sedans has a mean minimum time to travel a quarter mile of 15.4 seconds and a standard deviation of 2.12 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 } : \\)
\\( h _ { a } : \\)
(type integers or decimals. do not round.)
the claim is the hypothesis.
(b) use technology to find the p-value. find the standardized test statistic \\( t \\).
\\( t = \\)
(round to two decimal places as needed.)
obtain the p-value.
\\( p = \\)
(round to three decimal places as needed.)

Explanation:

Step1: Identify the null and alternative hypotheses

The null hypothesis \(H_0\) is a statement of equality. The alternative hypothesis \(H_a\) is the claim we are testing.
The claim is that the mean minimum time is greater than \(14.6\) seconds. So, \(H_0:\mu = 14.6\) and \(H_a:\mu>14.6\). The claim is the alternative hypothesis.

Step2: Calculate the test - statistic \(t\)

The formula for the \(t\) - statistic in a one - sample \(t\) - test is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\)
Given \(\bar{x} = 15.4\), \(\mu = 14.6\), \(s = 2.12\), \(n = 22\)

$$
LATEXBLOCK0
$$

Step3: Calculate the \(P\) - value

The degrees of freedom \(df=n - 1=22-1 = 21\). Since this is a right - tailed test (\(H_a:\mu>14.6\)), we use the \(t\) - distribution with \(df = 21\) to find the \(P\) - value.
Using a \(t\) - table or technology (e.g., TI - 84: tcdf(1.77,1E99,21)), the \(P\) - value is approximately \(0.045\)

Answer:

• \(H_0:\mu = 14.6\), \(H_a:\mu>14.6\), claim is \(H_a\)
• \(t\approx1.77\)
• \(P\approx0.045\)