Question

A report states that the cost of repairing a hybrid vehicle is falling even while typical repairs on conventional vehicles are getting more expensive. The most common hybrid​ repair, replacing the hybrid inverter​ assembly, had a mean repair cost of

​$3 comma 9273,927

in 2012. Industry experts suspect that the cost will continue to decrease given the increase in the number of technicians who have gained expertise on fixing​ gas-electric engines in recent months. Suppose a sample of

100100

hybrid inverter assembly repairs completed in the last month was selected. The sample mean repair cost was

​$3 comma 8603,860

with the sample standard deviation of

​$300300.

Complete parts​ (a) and​ (b) below.

a. Is there evidence that the population mean cost is less than

​$3 comma 9273,927​?

​(Use a

0.050.05

level of​ significance.)

State the null and alternative hypotheses.

H0​:

muμ

▼

not equals≠

greater than or equals≥

greater than>

less than or equals≤

less than<

equals=

​$nothing

H1​:

muμ

▼

equals=

greater than or equals≥

greater than>

less than<

less than or equals≤

not equals≠

​$nothing

​(Type integers or​ decimals.)

Find the test statistic for this hypothesis test.

test

statisticequals=nothing

​(Round to two decimal places as​ needed.)

The critical​ value(s) for the test statistic​ is(are)

nothing.

​(Round to two decimal places as needed. Use a comma to separate answers as​ needed.)

Is there sufficient evidence to reject the null hypothesis using

alphaαequals=0.050.05​?

A.

RejectReject

the null hypothesis. There is

sufficientsufficient

evidence at the

0.050.05

level of significance that the population mean cost is less than

​$3 comma 9273,927.

B.

Do not rejectDo not reject

the null hypothesis. There is

insufficientinsufficient

evidence at the

0.050.05

level of significance that the population mean cost is greater than

​$3 comma 9273,927.

C.

RejectReject

the null hypothesis. There is

sufficientsufficient

evidence at the

0.050.05

level of significance that the population mean cost is greater than

​$3 comma 9273,927.

D.

Do not rejectDo not reject

the null hypothesis. There is

insufficientinsufficient

evidence at the

0.050.05

level of significance that the population mean cost is less than

​$3 comma 9273,927.

b. Determine the​ p-value and interpret its meaning.

The​ p-value is

nothing.

​(Round to three decimal places as​ needed.)

What does this​ p-value mean given the results of part​ (a)?

A.The​ p-value is the probability that the actual mean cost is

​$3 comma 8603,860

or less.

B.The​ p-value is the probability that the actual mean cost is

​$3 comma 9273,927

or less given the sample mean cost is

​$3 comma 8603,860.

C.The​ p-value is the probability that the actual mean cost is more than

​$3 comma 8603,860.

D.The​ p-value is the probability of getting a sample mean cost of

​$3 comma 8603,860

or less if the actual mean cost is

​$3 comma 9273,927.

Answer 1

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ = 39273927

Ha: μ < 39273927

This corresponds to a left-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

(2) Rejection Region: Based on the information provided, the significance level is α=0.05, and the critical value for a left-tailed test is tc​=−1.66.

The rejection region for this left-tailed test is R=t:t<−1.66

(3) Test Statistics

The t-statistic is computed as follows:

tcritical value=-1.66

(4) Decision about the null hypothesis

CRITICAL VALUE APPROACH: Since it is observed that t=−2.95<tc​=−1.66, it is then concluded that the null hypothesis is rejected.Therefore, there is enough evidence to claim that the population mean μ is less than 3927, at the 0.05 significance level.

Using the P-value approach: The p-value is p=0.002, and since p=0.002<0.05, it is concluded that the null hypothesis is rejected. It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ is less than 3927, at the 0.05 significance level.

The P value is the probability that the actual mean cost is more than $ 3750.