Question

a used car dealer says that the mean price of a three - year - old sports utility vehicle is $19,000. you suspect this claim is incorrect and find that a random sample of 22 similar vehicles has a mean price of $18,774 and a standard deviation of $1572. assume the population is normally distributed. is there enough evidence to reject the claim at α = 0.05? complete parts (a) through (e) below. (a) write the claim mathematically and identify h₀ and hₐ. which of the following correctly states h₀ and hₐ? a. h₀: μ = $19,000; hₐ: μ ≠ $19,000 b. h₀: μ = $19,000; hₐ: μ = $19,000 c. h₀: μ > $19,000; hₐ: μ < $19,000 d. h₀: μ = $19,000; hₐ: μ > $19,000 e. h₀: μ ≥ $19,000; hₐ: μ < $19,000 f. h₀: μ ≠ $19,000; hₐ: μ = $19,000

Explanation:

In hypothesis testing, the null hypothesis \(H_0\) is a statement of equality. The claim here is about the mean price. The alternative hypothesis \(H_a\) is the statement we suspect. Since we are just checking if the claim (mean is \(\$19,000\)) is incorrect (a two - tailed test in the context of the problem, but looking at the options, the null hypothesis is the claim). The null hypothesis \(H_0\) is \(\mu=\$19,000\) and the alternative hypothesis \(H_a\) is \(\mu
eq\$19,000\). Looking at the options:

• Option A: \(H_0:\mu = 19000\), \(H_a:\mu

eq19000\) (assuming the symbol in \(H_a\) is a typo for \(
eq\)).

• Option B: \(H_0:\mu = 19000\), \(H_a:\mu = 19000\) is incorrect as both cannot be the same.
• Option C: \(H_0:\mu = 19000\), \(H_a:\mu\lt19000\) is a one - tailed test which is not indicated by the problem's "reject the claim" (a general rejection, not direction - specific).
• Option D: \(H_0:\mu = 19000\) is wrong as the value in \(H_0\) is incorrect.
• Option E: \(H_0:\mu\geq19000\) is not the claim (the claim is equality).
• Option F: \(H_0:\mu\gt19000\) is not the claim.

Answer:

A. \(H_0:\mu = \$19,000\), \(H_a:\mu
eq\$19,000\)