Question

A manufacturing process is designed to produce cylinder heads with a wall thickness that follows a normal distribution with a mean wall thickness of 0.22 inches and a standard deviation of 0.01 inches. The following table contains values of measured wall thicknesses produced by the process. Assume the standard deviation of the process is always 0.01 inches.

0.223, 0.228, 0.214, 0.193, 0.223, 0.213, 0.218, 0.233, 0.201, 0.223, 0.224, 0.231, 0.237, 0.204, 0.226, 0.219

a) What are appropriate null and alternative hypotheses for evaluating if the process is operating as designed?

b) If the rejection region is defined as mean < 0.215 or mean > 0.225 where the mean is from 16 samples, find the probability of a Type I error. Would this sample set lead to rejection of the null hypothesis?

c) What is the p-value for this sample?

d) What is an appropriate rejection region to make the probability of a Type I error equal to 1%?

e) Find the probability of a Type II error if the true mean is equal to the sample mean using the rejection region from part (b).

Answer 1

a)
Below are the null and alternate hypothesis
H0: mu = 0.22
Ha: mu not equals to 0.22

b)
P(X < 0.215)
= P(t < (0.215 - 0.22)/(0.01/sqrt(16)))
= P(t < -2)
= 0.0320

Type I error = 2*0.0320 = 0.0639

c)
Here, n = 16 , xbar = 0.2194 , mu = 0.22 and s = 0.01

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (0.2194 - 0.22)/(0.01/(sqrt(16))
t = -0.25

p-value = 0.806

d)
t-value = -2.9467
x = 0.22 -2.9467*0.01 = 0.1905
x = 0.22 +2.9467*0.01 = 0.2495

Rejection region < 0.1905 or > 0.2495