You have a sample of n = 9 observations from a normal distribution with = 1. You

Question

You have a sample of n = 9 observations from a normal distribution with = 1. You want to test

H0 : = 0 HA : > 0

where is the population mean. Suppose you use the following method as your test. You reject H0 if the sample mean is greater than 0 (y > 0) and you fail to reject H0 if the sample mean is less than or equal to 0 (y 0).

(a) Find the probability of a Type I error, that is, the probability that your test rejects H0 when in fact = 0.

(b) Find the probability of a Type II error when = 0.3. This is the probability that your test fails to reject H0 when in fact = 0.3

(c) Find the probability of a Type II error when = 1.

(d) Find the power of your test if = 0.3.

(e) Find the power of your test if = 1.

Explanation / Answer

given n=9

= 1

consider the following table

So, a type I error is rejecting H0 when H0 is true, like sending an innocent person to prison
a type II error is letting a guilty person go free after the trial.

P(Type I Error)

P(Type II Error) =

We generally don't work with Type II errors and instead talk about Power

Power = 1 - P(Type II Error) = 1 -

in developing tests we try to maximize the Power and minimize .

A) we have a 50% probability of a type I error

B) P(Type II Error) = P( X < 0)

where X ~ Normal( mean = 0.3, variance = 1/9)

P(X < 0) = P( Z < (0 - 0.3) / (sqrt(1/9)))   = 0.1841

C) P(X < 0) when X ~ Normal(mean = 1, variance = 1/9)

P(X < 0) = P(Z < (0 - 1) / sqrt(1/9)) = 0.00135

D) Power = 1- 0.1841 = 0.8159

E) Power = 1- 0.00135 = 0.99865

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