Question

The manufacturer of a sports car claims that the fuel injection system lasts 48 months before it needs to be replaced. A consumer group tests this claim by surveying a random sample of 10 owners who had the fuel injection system replaced.

The ages of the cars at the time of replacement were (in months): 23 42 49 48 53 46 30 51 42 52

Use the sample data to calculate the mean age of a car when the fuel injection system fails and the standard deviation. (Round your answers to two decimal places.)

x = months s = months Test the claim that the fuel injection system lasts less than an average of 48 months before needing replacement. Use a 5% level of significance.

a.) What are we testing in this problem?

-single mean

-single proportion

b.) What is the level of significance? _______

c.) State the null and alternate hypotheses.

-H0: μ ≥ 48; H1: μ < 48

-H0: μ ≤ 48; H1: μ > 48

-H0: p ≥ 48; H1: p < 48

-H0: p = 48; H1: p ≠ 48

-H0: p ≤ 48; H1: p > 48

-H0: μ = 48; H1: μ ≠ 48

d.) What sampling distribution will you use? What assumptions are you making?

-The Student's t, since we assume that x has a normal distribution with unknown σ.

-The Student's t, since we assume that x has a normal distribution with known σ.

-The standard normal, since we assume that x has a normal distribution with unknown σ.

-The standard normal, since we assume that x has a normal distribution with known σ.

e.) What is the value of the sample test statistic? (Round your answer to three decimal places.) ______

f.) Estimate the P-value.

-P-value > 0.250

-0.125 < P-value < 0.250

-0.050 < P-value < 0.125

-0.025 < P-value < 0.050

-0.005 < P-value < 0.025

-P-value < 0.005

g.) Sketch the sampling distribution and show the area corresponding to the P-value. ______

h.) Will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

--At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

-At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

-At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

-At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

i.) Interpret your conclusion in the context of the application.

-There is sufficient evidence at the 0.05 level to conclude that the injection system lasts less than an average of 48 months.

-There is insufficient evidence at the 0.05 level to conclude that the injection system lasts less than an average of 48 months.

Answer 1

A)

-single mean

b)

0.05

c)

-H0: μ ≥ 48; H1: μ < 48

d)

The Student's t, since we assume that x has a normal distribution with unknown σ.

e)

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (43.6 - 48)/(9.9017/sqrt(10))
t = -1.405

f)

P-value = 0.0968
-0.050 < P-value < 0.125

h)

-At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

i_

There is insufficient evidence at the 0.05 level to conclude that the injection system lasts less than an average of 48 months.