Probability of making a Type II Error

To find out the probability of making a type II error, let’s see an example, suppose we have hypotheses such as


Ho: μ0 = 500 

(null hypothesis with specific population mean)


Ha: μ > 500 

(alternative hypothesis with an assumption that

the population mean could be greater than μ0 )


for a sample size of n = 40

with population standard deviation (σ) of 115

at the level of significance α that is probability of making type I error is 0.01


Find the probability of making type II error if the population mean is μ = 524.



first we need to find out from the data what are


the specific value of the population mean (μ0) given in the null hypothesis (H0),

level of significance (α),

standard deviation of the population (σ)

the sample size (n), and

population mean μ.


In this example, they are

μ0 = 500

α = 0.01

σ = 115

n = 40

μ = 524


From the level of significance (α), calculate z score


for two-tail test, use α/2 to find z score

for one-tail test, use α to find z score


e.g.

if α= 0.05, then use 0.025 for two-tail test

if α= 0.05, then use 0.05 for one-tail test


But most of the time, we just read it out of the α- table (see table)

Level of Significance


0.10 (10%)


0.05 (5%)


0.01 (1%)

One-Tail Test


1.28


1.645


2.33



Use + for right-tail

Use - for left-tail

Two-Tail Test


1.645


1.96


2.575



Use ± for two-tail

In this example, α= 0.05, and it is a one-tail test, see Ha: μ > 500 

then from the α- table, use the value +2.33,

2.33 is + because it is a right-tail test (the sign > pointing to the right)


Then find sample mean (x bar)


Use       x bar = μ0 ± zα/2 . σ/√n     for two-tail test


Use       x bar = μ0 ± zα   . σ/√n     for one-tail test, for right use +, for left use -


In this example, it is a one-tail test (right-tail, so it is +)


x bar = μ0 + zα   . σ/√n = 500 + [+2.33 * (115/√40) ] = 542



After getting the sample mean x bar,

use it to find the z score in the following formula


Z = (x bar - μ)/(σ/√n )


where μ is the population mean, do not get confuse with the other population mean (μ0) mentioned in the null hypothesis (H0).

They are different.


In this example,


Z542 = (x bar - μ)/(σ/√n ) = (542 - 524)/(115/√40) = 0.9899



Then use this Z value to compute the probability of Type II Error based on the interval of the population mean stated in the alternative hypothesis.


In this example:

Ho: μ0 = 500 

Ha: μ > 500

μ = 524


Draw a normal curve with population mean μ = 524, and sample mean found which is x bar = 542


The normal curve shows shaded area that is less than x bar = 542

This shaded area describes the probability of type II error


Remember the x bar = 542 is calculated from

the level of significance α = 0.01,

and α is actually the probability of type I error.


Remember by reducing the probability of type I error,

we are increasing the probability of making type II error.


P (Type II Error) = P ( Z < Z542 ) = P ( Z < 0.9899 ) = 0.8389


EXCEL:

NORMSDIST(0.9899) = 0.8389


Therefore, the probability of type II error, in this example is 0.8389.


P (Type II Error) = β

P (Type I Error) = level of significance = α

The consequence of a small α is large β.


Be careful, (1-β) is not α

because (1-β) = the power of the test.