Question

a car company says that the mean gas mileage for its luxury sedan is at least 25 miles per gallon (mpg). you believe the claim is incorrect and find that a random sample of 5 cars has a mean gas mileage of 23 mpg and a standard deviation of 2 mpg. at \\( \alpha = 0.05 \\), test the companys claim. assume the population is normally distributed. click here to view the t - distribution table. click here to view page 1 of the normal table. click here to view page 2 of the normal table. which sampling distribution should be used and why? a. use a normal sampling distribution because the population is normal, and \\( \sigma \\) is unknown. b. use a t - sampling distribution because \\( n<30 \\). c. use a normal sampling distribution because \\( n > 30 \\). d. use a normal sampling distribution because the population is normal, and \\( \sigma \\) is known. e. use a t - sampling distribution because the population is normal, and \\( \sigma \\) is known. f. use a t - sampling distribution because the population is normal, and \\( \sigma \\) is unknown. state the appropriate hypotheses to test. a. \\( h_{0}: \mu \leq 25 \\) \\( h_{a}: \mu>25 \\) b. \\( h_{0}: \mu \
eq 25 \\) \\( h_{a}: \mu = 25 \\) c. \\( h_{0}: \mu \geq 25 \\) \\( h_{a}: \mu<25 \\) d. \\( h_{0}: \mu = 25 \\) \\( h_{a}: \mu \
eq 25 \\)

Explanation:

Step1: Determine the sampling distribution

Since the population is normally distributed, \(n = 5<30\), and \(\sigma\) (population standard deviation) is unknown (we are given the sample standard deviation \(s = 2\)), we use the t - sampling distribution.

Step2: State the hypotheses

The company claims that \(\mu\geq25\). The null hypothesis \(H_{0}\) is the claim we assume to be true for testing purposes. The alternative hypothesis \(H_{a}\) is what we try to find evidence for. If we believe the claim (\(\mu\geq25\)) is incorrect, and we suspect \(\mu < 25\) (because the sample mean \(23<25\)), then \(H_{0}:\mu\geq25\) and \(H_{a}:\mu < 25\)

Answer:

For the sampling distribution: F. Use a t - sampling distribution because the population is normal, and \(\sigma\) is unknown.
For the hypotheses: C. \(H_{0}:\mu\geq25\), \(H_{a}:\mu < 25\)