Question

Problem 2 You’re developing a foam insulation product that expands after application. You want the final (after expansion) height of the insulation to be more than 175 mm. You performed an experiment of spraying the insulation on 10 different surfaces (in a similar manner on similar surfaces), and you recorded the following insulation heights. 171.9 186.2 195.6 164.6 147.8 155.4 191.5 201.2 171.4 201.1

a. Is there sufficient evidence to conclude that your condition is met? Use the critical value approach with significance of 0.1.

b. Find the probability of type II error if the true mean height is 185 mm.

c. Find the probability of type II error if the true mean height is 195 mm.

d. Discuss how and why the probability of type II error changes.

Answer 1

First we need to find the mean and SD of data. Following is the output of descriptive statistics:

Descriptive statistics
	X
count	10
mean	178.670
sample standard deviation	19.177
sample variance	367.749
minimum	147.8
maximum	201.2
range	53.4

(a)

(b)

First we need to find the critical value of sample mean for which we will reject the null hypothesis.

The critical value of sample mean is

The t -score for and is

The type II error will be

Excel function used to find the above probability is : "=TDIST(0.27,9,1)"

(c)

First we need to find the critical value of sample mean for which we will reject the null hypothesis.

The critical value of sample mean is

The t -score for and is

The type II error will be

Excel function used to find the above probability is : "=TDIST(1.91,9,1)"

(d)

As the true population mean changes, or type I error changes sample size changes type II error will change.