Question

A box contains a large number of chips of two different colors: red and blue, and it is desired to test the null hypothesis (H0) that chips of the two colors are in equal proportions against the alternative hypothesis (H1) that they are not in equal proportions. Suppose that four chips are to be drawn at random from the box, and H0 is to be rejected if and only if at least three of the chips have the same color.

(a) Determine the probability of committing type I error.

(b) Determine the probability of committing type II error if 1/3 of the chips are red.

Answer 1

a) The probability of type I error is computed as the probability of rejecting a true null hypothesis. Therefore here it would be conducted as the probability of getting 3 or 4 chips of the same colour when the true proportion of each of the two colours is 0.5. Therefore the probability here is computed as:

= 1 - Probability that 2 chips each of the two colours is taken out

= 1 - (4c2)*0.5⁴ = 0.625

Therefore 0.625 is the required probability here.

b) The probability of committing a type II error if the true proportion of the red chips is 1/3 is computed here as:

= Probability of getting 2 chips of each colour when the probability of getting a red chip is in fact 1/3

= (4c2)*(1/3)²*(2/3)² = 0.2963

Therefore 0.2963 is the required probability here.