Question

question 12, 7.3.19 - t
part 2 of 6
hw score: 36.59%, 8.04 of 22 points
points: 0 of 1
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 $19,774 and a standard deviation of $1972. is there enough evidence to reject the claim at α = 0.05? complete parts (a) through (e) below
assume the population is normally distributed.
(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
(b) find the critical value(s) and identify the rejection region(s).
what is(are) the critical value(s), t₀?
t₀ =
(use a comma to separate answers as needed. round to three decimal places as needed.)

Explanation:

Step1: Identify the null and alternative hypotheses

The claim is that the mean price is \( \mu = 19000\). The null hypothesis \(H_0\) is the statement of equality, so \(H_0:\mu = 19000\). The alternative hypothesis \(H_a\) is the statement we are trying to find evidence for. Since we suspect the claim is incorrect (a two - tailed test), \(H_a:\mu
eq19000\).

Step2: Determine the critical values

Since the population standard deviation \(\sigma\) is unknown and \(n = 22\) (small sample, \(n<30\)), we use the \(t\) - distribution. The degrees of freedom \(df=n - 1=22-1 = 21\). For a two - tailed test with \(\alpha = 0.05\), the critical values are \(t_{\alpha/2}\) and \(-t_{\alpha/2}\).
Using a \(t\) - table or calculator, \(t_{0.025,21}= 2.080\) and \(-t_{0.025,21}=- 2.080\). The rejection regions are \(t>2.080\) and \(t < - 2.080\).

Answer:

(a) \(H_0:\mu = 19000\), \(H_a:\mu
eq19000\) (the first option in the given multiple - choice for part (a)).
(b) The critical values are \(t = 2.080\) and \(t=-2.080\). The rejection regions are \(t>2.080\) and \(t < - 2.080\).