Question

assume that the average gas mileage of all automobiles is 21.5 miles per gallon. for a random sample of 50 sport utility vehicles (suvs), the mean gas mileage is 21.3 miles per gallon. assume a population standard deviation of 3.5 miles per gallon. test the claim that the mean mileage of all suvs is less than 21.5 miles per gallon. use a 0.05 significance level.
state the null and alternative hypotheses.
a. ( h_0: mu = 21.5 ) miles per gallon
( h_a: mu < 21.5 ) miles per gallon
b. ( h_0: mu = 21.5 ) miles per gallon
( h_a: mu > 21.5 ) miles per gallon
c. ( h_0: mu
eq 21.5 ) miles per gallon
( h_a: mu = 21.5 ) miles per gallon
d. ( h_0: mu = 21.5 ) miles per gallon
( h_a: mu
eq 21.5 ) miles per gallon
find the test statistic.
test statistic = (round to two decimal places as needed.)
find the p - value.
the p - value is
(round to four decimal places as needed.)
state the conclusion.

Explanation:

Step1: State the hypotheses

The null hypothesis \(H_0\) is a statement of no - effect or equality. The alternative hypothesis \(H_a\) is the claim we are testing. Since we are testing the claim that the mean mileage of all SUVs is less than \(21.5\) miles per gallon, \(H_0:\mu = 21.5\) miles per gallon and \(H_a:\mu<21.5\) miles per gallon. So the correct hypothesis set is Option A.

Step2: Calculate the test statistic

The formula for the z - test statistic when the population standard deviation \(\sigma\) is known is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). Here, we assume \(\bar{x} = 21.3\) (given average gas - mileage of all automobiles, which we can use as the sample mean for the test, \(\mu = 21.5\), \(\sigma=3.5\), and \(n = 50\).

$$
LATEXBLOCK0
$$

Step3: Calculate the P - value

For a one - tailed (left - tailed) z - test, the P - value is \(P(Z<z)\). Using a standard normal table or a calculator with a normalcdf function (\(normalcdf(-\infty,-0.40)\)), we get \(P - value=\Phi(-0.40)\)

$$P - value=0.3446$$

Answer:

• Hypotheses: A. \(H_0:\mu = 21.5\) miles per gallon, \(H_a:\mu<21.5\) miles per gallon
• Test statistic: \(-0.40\)
• P - value: \(0.3446\)
• Conclusion: Since the P - value (\(0.3446\))> significance level (\(0.05\)), we fail to reject the null hypothesis. There is not enough evidence at the \(0.05\) significance level to support the claim that the mean mileage of all SUVs is less than \(21.5\) miles per gallon.