# statistics problem regarding null and alternative hypotheses?

you want to see if a redesign of the cover of a mail-order catalog will increase sales. a very a large # of customers will receive the original catalog, and a random sample of customers will receive the one with the new order. for planning purposes, you are willing to assume that the mean sales from the catalog will be approx. Normal with standard deviation=50dollars and the the mean for the original catalog will be mu=25 dollars. you decide to use n=900. you wish to test null=25 and alternative>25. find the probability of a type I error, that is, the prob. that your test rejects the null when in fact mu=25dollars.

Relevance

The probability of a Type I error is bounded by the significance level of the test. You did not define a significance level or a rejection region here. there is not enough information given to find the probability of a Type I Error.

Let α be the significance level of the test

consider the following table

_ _ _ _ _ _ Reject H0 _ _ _ _ Fail to Reject H0

H0 is true _ Type I error _ _ _ _ _ ☺ _ _ _

H0 is false _ _ _ ☺ _ _ _ _ _ _ Type II error _

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) ≤ α

P(Type II Error) = β

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

Power = 1 - P(Type II Error) = 1 - β

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