Question

a car company says that the mean gas mileage for its luxury sedan is at least 23 miles per gallon (mpg). you believe the claim is incorrect and find that a random sample of 6 cars has a mean gas mileage of 21 mpg and a standard deviation of 3 mpg. at \\( \alpha = 0.10 \\), 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 t - sampling distribution because \\( n<30 \\). b. use a normal sampling distribution because \\( n > 30 \\). c. use a t - sampling distribution because the population is normal, and \\( \sigma \\) is unknown. d. use a normal sampling distribution because the population is normal, and \\( \sigma \\) is unknown. e. use a t - sampling distribution because the population is normal, and \\( \sigma \\) is known. f. use a normal sampling distribution because the population is normal, and \\( \sigma \\) is known. state the appropriate hypotheses to test. a. \\( h_0: \mu\geq23 \\) \\( h_a: \mu < 23 \\) b. \\( h_0: \mu = 23 \\) \\( h_a: \mu\
eq23 \\) c. \\( h_0: \mu\leq23 \\) \\( h_a: \mu > 23 \\) d. \\( h_0: \mu\
eq23 \\) \\( h_a: \mu = 23 \\)

Explanation:

For the sampling distribution question:

• The t - sampling distribution is used when the population is normal, the sample size \(n\) is small (\(n<30\)), and the population standard deviation \(\sigma\) is unknown. Here, \(n = 6<30\), the population is normal, and we are given the sample standard deviation \(s\) (since \(\sigma\) is unknown).

For the hypothesis question:

• The company claims that \(\mu\geq23\). The null hypothesis \(H_0\) is the statement being tested (the claim), and the alternative hypothesis \(H_a\) is the statement we suspect. Since we believe the claim (\(\mu\geq23\)) is incorrect, the alternative hypothesis is \(\mu < 23\).

Answer:

For the sampling distribution: C. Use a t - sampling distribution because the population is normal, and \(\sigma\) is unknown.
For the hypotheses: A. \(H_0:\mu\geq23\), \(H_a:\mu < 23\)