Question

Tanner loves to go hiking. He has gone on 40 hikes over the last two years, with an average distance of 12.8 miles and a standard deviation of 1.4 miles.

a) Based on the information provided, do you need to apply the Central Limit Theorem, and can you apply it?

b) Construct a 95% confidence interval for the average distance of the hikes Tanner takes.

c) Tanner’s wife claims he always goes on hikes that are too long. He asks her what she means by “too long” and she clarifies that as more than 10 miles. Tanner decides to test to see if this is a reasonable claim at a significance level of 0.05. What are the hypotheses for this test?

d) Calculate the t-test statistic for the test defined in part (c).

e) Find the P-value for the test Tanner is conducting.

f) Make your conclusion for the hypothesis test and interpret its meaning in the context of the problem.

Answer 1

x̅ = 12.8, s = 1.4, n = 40

a) Yes, we can Central Limit Theorem as the sample size is more than 30.

b) 95% Confidence interval :

At α = 0.05 and df = n-1 = 39, two tailed critical value, t-crit = T.INV.2T(0.05, 39) = 2.023

Lower Bound = x̅ - t-crit*s/√n = 12.8 - 2.023 * 1.4/√40 = 12.352

Upper Bound = x̅ + t-crit*s/√n = 12.8 + 2.023 * 1.4/√40 = 13.248

12.352 < µ < 13.248

c) Null and Alternative hypothesis:

Ho : µ = 10

H1 : µ > 10

d) Test statistic:

t = (x̅- µ)/(s/√n) = (12.8 - 10)/(1.4/√40) = 12.6491

e) df = n-1 = 39

p-value = T.DIST.RT(12.6491, 39) = 0.0000

f) Decision:

p-value < α, Reject the null hypothesis

Conclusion:

There is enough evidence to conclude that the mean of hikes is more than 10 at 0.05 significance level.