Question

The average annual miles driven per vehicle in the United States is 11.1 thousand miles, with σ ≈ 600 miles. Suppose that a random sample of 41 vehicles owned by residents of Chicago showed that the average mileage driven last year was 10.9 thousand miles. Does this indicate that the average miles driven per vehicle in Chicago is different from (higher or lower than) the national average? Use a 0.05 level of significance.

What are we testing in this problem?

single proportion

single mean

(a) What is the level of significance?

State the null and alternate hypotheses.

H0: μ = 11.1; H1: μ ≠ 11.1

H0: p = 11.1; H1: p > 11.1

H0: p = 11.1; H1: p ≠ 11.1

H0: μ = 11.1; H1: μ < 11.1

H0: μ = 11.1; H1: μ > 11.1

H0: p = 11.1; H1: p < 11.1

(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 unknown σ.

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

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

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

P-value > 0.500

0.250 < P-value < 0.500

0.100 < P-value < 0.250

0.050 < P-value < 0.100

0.010 < P-value < 0.050

P-value < 0.010

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

(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 miles driven per vehicle in the city differs from the national average.

There is insufficient evidence at the 0.05 level to conclude that the miles driven per vehicle in the city differs from the national average.

Answer 1

Given that, population standard deviation σ ≈ 600 miles

=>  σ ≈ 600/1000 = 0.6 thousand miles

sample size (n) = 41

sample mean $(\bar x)$ = 10.9 thousand miles

a) level of significance is α = 0.05

The null and alternative hypotheses are,

H0: μ = 11.1; H1: μ ≠ 11.1

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

Test statistic is,

$Z = \frac {10.9-11.1}{\frac {0.6}{\sqrt {41}}} = -2.13$

Test statistic is, z = -2.13

c) p-value = 2 * P(Z < -2.13)= 2 * 0.0166 = 0.0332

0.010 < P-value < 0.050

d) Since, p-value = 0.0322 < 0.05

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

e) Conclusion : There is sufficient evidence at the 0.05 level to conclude that the miles driven per vehicle in the city differs from the national average.