Question

A bag contains 20 envelopes, all the same size but some have $1 in them and some have $100. You are told to pick envelopes one at a time with replacement . You keep picking the envelopes in this way until you get an envelope that has $100. Let Y = the number of envelopes picked in order to get one with $100. Test if the proportion of envelopes in the bag with $100 is over 0.10. It is decided to use a rejection region of Y <3.

a)Find the probability of a Type I error if you use this test.

b)Find the probability of a Type II error if the bag really contains 6 envelopes with $100

Answer 1

Here we're testing H₀: p = 0.1 vs H_a: p>0.1

a) We know that the probability of a Type I error is defined as the probability of rejecting the null hypothesis when that is true.

The probability of type I error is given by:

b) We know that the probability of a Type II error is defined as the probability of accepting the null hypothesis when it is false. Here given that the bag really contains 6 envelopes with $100, thus p=0.3. Thus the probability of type II error is given by:

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