Question

a car company says that the mean gas mileage for its luxury sedan is at least 23 miles per gallon (mpg). you believe this claim is incorrect and find that a random sample of 7 cars has a mean gas mileage of 20 mpg and a standard deviation of 4 mpg. at α = 0.025, test the companys claim. assume the population is normally distributed
which sampling distribution should be used and why?
a. use a t - sampling distribution because the population is normal, and σ is known
b. use a normal sampling distribution because the population is normal and σ is known
c. use a t - sampling distribution because n < 30
d. use a normal sampling distribution because n > 30
e. use a normal sampling distribution because the population is normal, and σ is unknown
f. use a t - sampling distribution because the population is normal and σ is unknown
state the appropriate hypotheses to test.
a. h₀: μ ≥ 23
hₐ: μ < 23
b. h₀: μ ≤ 23
hₐ: μ > 23
c. h₀: μ = 23
hₐ: μ ≠ 23
d. h₀: μ ≠ 23
hₐ: μ = 23

Explanation:

• For the sampling distribution: When the population is normally distributed and the population standard deviation \(\sigma\) is unknown (here, we are given the sample standard deviation \(s = 4\)), we use the \(t -\)sampling distribution. The formula for the \(t -\)test statistic is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) where \(\bar{x}\) is the sample mean, \(\mu\) is the hypothesized population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.
• For the hypotheses: The company claims that \(\mu\geq23\). The null hypothesis \(H_0\) is the statement of equality or the claim we assume to be true for the purpose of testing. The alternative hypothesis \(H_a\) is the statement we are trying to find evidence for. Since we believe the claim (\(\mu\geq23\)) is incorrect and we want to test if the mean is less than 23, \(H_0:\mu = 23\) (a common form when testing against a one - sided alternative for a claim of “at least” in the context of hypothesis testing for the mean when the claim is about the population mean) and \(H_a:\mu<23\)

Answer:

• Sampling distribution: Use a \(t -\)sampling distribution because the population is normal and \(\sigma\) is unknown.
• Hypotheses: \(H_0:\mu = 23\), \(H_a:\mu<23\) (corresponds to option A for the hypotheses if we assume the options are ordered as in a typical multiple - choice setup where \(H_0:\mu\geq23\) can be tested as \(H_0:\mu = 23\) against \(H_a:\mu<23\) in practice for one - sided \(t -\)tests)