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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):

27 | 44 | 43 | 48 | 53 | 46 | 30 | 51 | 42 | 52 |

(i) Use your calculator to calculate the mean age of a car when the fuel injection system fails *x* and standard deviation *s*. (Round your answers to two decimal places.)

x = | months |

s = | months |

(ii) 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.

What are we testing in this problem?

single meansingle proportion

(a) What is the level of significance?

State the null and alternate hypotheses.

*H*_{0}: *μ* = 48; *H*_{1}: *μ* ≠ 48*H*_{0}: *p* = 48; *H*_{1}: *p* ≠ 48 *H*_{0}: *p* = 48; *H*_{1}: *p* < 48*H*_{0}: *μ* = 48; *H*_{1}: *μ* < 48*H*_{0}: *μ* = 48; *H*_{1}: *μ* > 48*H*_{0}: *p* = 48; *H*_{1}: *p* > 48

(b) What sampling distribution will you use? What assumptions are you making?

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 known *σ*. The standard normal, 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 unknown *σ*.

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

(c) Find (or estimate) the *P*-value.

*P*-value > 0.2500.125 < *P*-value < 0.250 0.050 < *P*-value < 0.1250.025 < *P*-value < 0.0500.005 < *P*-value < 0.025*P*-value < 0.005

Sketch the sampling distribution and show the area corresponding to the *P*-value.

(d) Based on your answers in parts (a) to (c), 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.

(e) 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.

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