Question

If we were to conduct a hypothesis test to test if a population proportion of a certain event is 0.2 versus the alternative that it is greater than 0.2 and we sampled 100 people, what would the probability of a Type I Error be if we were to use the arbitrary decision rule to reject Ho if more than 30 units in the sample had the event (not the way we tested hypotheses in class)? What would the probability of a Type II Error be with the same decision rule if the real proportion was 0.25?

Answer 1

Claim : Population proportion of a certain event is greater than 0.2

: P = 0.2 vs : P > 0.2  

= 0.2 and = = = 0.04

P ( Type I error ) = P( reject H0 , when H0 is true )

= P( reject H0 , if > 30/100 )

= P( > 0.3 )

=

= P( z > 2.5 )

= 1 - P( z ≤ 2.5 )

= 1 - 0.9938

= 0.0062

Probability of a Type I Error is 0.0062

For real proportion P = 0.25

= 0.25 and = = = 0.0433

Probability of a Type II Error = P ( Fail to reject H0, when H0 is false )

= P(    ≤ 0.3 , when P = 0.25 )

=

= P( z ≤ 1.15 )

= 0.8749

Probability of a Type II Error is 0.8749