Question

A machine is required to produce bolts with an average length of 2.00 cm. A random sample of 120 bolts had a mean length of 1.991 cm and a standard deviation of 0.766 cm.

1. (a) [7] Does the sample provide sufficient evidence that the machine is not operating properly? Test an appropriate hypothesis at α = 0.01.

2. (b) [5] Make a 99% confidence interval for the average length. What additional information does it provide over the test in (a)?

3. (c) If the sample does not come from a normal distribution how could the use of the procedures in (a) & (b) be justified?

Answer 1

a)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 2
Alternative Hypothesis, Ha: μ ≠ 2

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (1.991 - 2)/(0.766/sqrt(120))
t = -0.129

P-value Approach
P-value = 0.8976
As P-value >= 0.01, fail to reject null hypothesis.

There is not sufficient evidence that the machine is not operating properly

b)
sample mean, xbar = 1.991
sample standard deviation, s = 0.766
sample size, n = 120
degrees of freedom, df = n - 1 = 119

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, tc = t(α/2, df) = 2.618

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (1.991 - 2.618 * 0.766/sqrt(120) , 1.991 + 2.618 * 0.766/sqrt(120))
CI = (1.808 , 2.174)

c)
The sample size is greater than 120 hence we can apply central limit theorem.