Question

1. A research engineer for a tire manufacturer is investigating tire life for a new rubber compound. She has built 10 tires and tested them to end-of-life in a road test. The sample mean and standard deviation are 61,492 and 3035 kilometers, respectively.  

(a) Can you conclude, using α = 0.05, that the standard deviation of tire life exceeds 3000 km?

(b) Find the P-value for this test.

(c) Find a 95% lower confidence bound for σ and use it to test the hypothesis.

Answer 1

a)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: σ = 3000
Alternative Hypothesis, Ha: σ > 3000

Test statistic,
Χ^2 = (n-1)*s^2/σ^2
Χ^2 = (10 - 1)*3035^2/3000^2
Χ^2 = 9.211

P-value Approach
P-value = 0.418

As P-value >= 0.01, fail to reject null hypothesis.
No, the std. dev. does not exceed 3000km

c)
α = 1 - 0.95 = 0.05
The critical values for α = 0.05 and df = 9 are Χ^2(1-α/2,n-1) = 2.7 and Χ^2(α/2,n-1) = 19.023

CI = (sqrt(9*3035^2/19.023) , sqrt(9*3035^2/2.7))
CI = (2087.57 , 5541.13)

lower confidence bound = 2087.57