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):

21

42

43

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 proportion

single mean     

b. What is the level of significance?


c. State the null and alternate hypotheses.

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

H₀: μ ≥ 48; H₁:  μ < 48

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

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.     

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.

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

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 =	42.8 months
s =	10.18 months

a. What are we testing in this problem?

single mean ( since mean is compared )

b. What is the level of significance?

5% = 0.05

c. State the null and alternate hypotheses.

H₀: μ ≤ 48; H₁:  μ > 48 ( since claim is that it is more than 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.

e. What is the value of the sample test statistic?

t = 1.615.

f. Estimate the P-value.

df = n - 1 = 10

right tailed test

p-value = 0.0704

0.050 < P-value < 0.125

.

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 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.