Question

The true population mean weight of a manufactured widget is 178 grams.

A hypothesis test is designed to determine if the mean weight (mu) is less than 180 grams.

Ho: mu >= 180

Ha: mu < 180

What is the probability of Type II error if you are requiring 90% confidence of a sampling process includes 100 observations with a population standard deviation of 12?

Answer 1

Formula for Type II Error Rate:

Given the mean weight of a manufactured widget is 180 gm. The true population mean weight of a manufactured widget is 178 gm, and the population standard deviation is 12. At .0.01 significance level, what is the probability of having type II error for a sample size of 100

Given,

H₀ (?₀) = 180, H_A (?_A) = 178, ? = 12, n = 100

To Find,

Beta or Type II Error rate

Solution:

Step 1:

Let us first calculate the value of c, Substitute the values of H0, HA, ? and n in the formula,

(c - ?0) / (? / ?n)	= -1.28
(c - 180) / (12 / ?(100))	= -1.28
c - 180	= -1.536
c	= 178.464

Step 2:

In the formula, take ? to the left hand side and the other values to right hand side, ? = 1 - p(z > (c - ?_A / (? / ?n))) Here,[ z =x - ?_A / (? / ?n) ] Substitute the values in the above equation,

?	= 1 - p(z > ((178.464 - 178) / (12 / ?(100))))
	= 1 - p(z > 0.39)
	= 1 - 0.3483
	= 0.6517

Hence the Type II Error rate value is calculated.