# Type II error calculation

#### PBRN

##### New Member
My question: It is desired to test H0: μ = 55 against H1: μ < 55 using α = 0.10. The population in question is uniformly distributed with a standard deviation of 20. A random sample of 64 will be drawn from this population. If μ is really equal to 50, what is the probability that the hypothesis test would lead the investigator to commit a Type II error?

What I have done: read statistics text (Statistics - informed decisions using data) and the posted online learning by the instructor, multiple times. My calculations:

Left Tail Test: 55 - 2.575 (20/ sq of 64: 8) = 48.5625

P (Type II error): (50 - 48.5625) / (20/ sq of 64: 8) = 0.575

This should equal .7157 probability of a Type II error

I know this is not the correct answer. I just can't figure out what I am doing wrong. any articles or references or advise on here would be much appreciated.

#### Dragan

##### Super Moderator
My question:

Left Tail Test: 55 - 2.575 (20/ sq of 64: 8) = 48.5625
Where on earth did you get this idea from?

#### PBRN

##### New Member
Where on earth did you get this idea from?
The inequality formula in the text is: x = pop. mean minus critical value times (pop. standard deviation divided by square root of sample size)

The probability if type II error in the text is: result of inequality minus true pop. mean divided by (pop. standard deviation divided by square root of sample size)

then using standard normal distribution table to get the probability

Again - maybe I am way off base - that is why I posted.

#### Dragan

##### Super Moderator
The inequality formula in the text is: x = pop. mean minus critical value times (pop. standard deviation divided by square root of sample size)

The probability if type II error in the text is: result of inequality minus true pop. mean divided by (pop. standard deviation divided by square root of sample size)

then using standard normal distribution table to get the probability

Again - maybe I am way off base - that is why I posted.
PBRN: Why are you using a critical value of 2.575 when the Type I error rate is specified to be 0.10?--(look up at your first post)

#### PBRN

##### New Member
I thought 0.1 = 2.575 and 0.05 = 1.645 for critical values.

#### Dragan

##### Super Moderator
I thought 0.1 = 2.575 and 0.05 = 1.645 for critical values.
No, that is not correct.

Remember, you're only considering the left-hand side of the distribution because the hypothesis is directional.

The appropriate Critical value is -1.28155 for a specified alpha rate of 0.1 (when your considering the left-hand side of the normal distribution)

And, by the way, in your first post, the value of 0.575 that you computed is not a probability--it's a Z-score.

#### PBRN

##### New Member
Thanks for the help:

55 - 1.2875 (2.5) = 51.78125

(50 - 51.78125)/2.5 = -0.7125

-.7125 = .2358

again thanks, I spent hours going round and roung with this question.