Question

1. 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 \\) 4 14.8
\\( h _ { a } : \mu \\) 4 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 = \square \\)
(round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for the t - test statistic

The formula for the t - test statistic in a one - sample t - test is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean under the null hypothesis, \(s\) is the sample standard deviation, and \(n\) is the sample size.
We are given that \(\bar{x} = 15.5\), \(\mu=14.8\), \(s = 2.12\), and \(n = 21\).

Step2: Substitute the values into the formula

$$
LATEXBLOCK0
$$

First, calculate \(\sqrt{21}\approx4.5837\), then \(2.12/\sqrt{21}\approx2.12\div4.5837\approx0.4625\)

$$ t=\frac{0.7}{0.4625}\approx1.51 $$

Answer:

\(t\approx1.51\)