Question

I have a wheel that is divided into five equal pie-slice shaped wedges, each of which is a different color (red, blue, green, yellow, purple). The wheel has an arrow at the center, and we can spin the wheel and see what color it ends on when the wheel stops spinning. I want to test that the wheel is fair (each color is equally likely) by spinning the arrow over and over again, independently, until the second time I see any particular color. I will conclude that the wheel is not fair if it takes three or fewer rolls for a particular color to come up two times.

Q1) The significance level of the test is: A) 10% B) 30% C) 50% D) 70% E) 90%

Q2) The power of this test against the alternative hypothesis that the chance the arrow ends on purple is 52% and the chance it ends each of the other four colors is 12% is closest to A) 10% B) 30% C) 50% D) 70% E) 90%

Answer 1

Solution

Back-up Theory

Significance level α of a test = P(Type I Error) ......................................................................................... (1)

Type I Error is the error of rejecting a null hypothesis when it is true........................................................(1a)

α = P(Type I Error) = probability of rejecting a null hypothesis when it is true........................................ (1b)

Thus, Significance level α of a test = probability of rejecting a null hypothesis when it is true................ (1c)

Type II Error is the error of accepting a null hypothesis when it is not true, i.e., Alternative is true.......... (2)

β = P(Type II Error)

= probability of accepting a null hypothesis when it is not true, i.e., Alternative is true........................... (2a)

Power of a test = 1 – β

= probability of rejecting a null hypothesis when it is not true, i.e., Alternative is true............................ (2b)

Or equivalently, power is the probability of accepting a true alternative. ................................................... (2c)

Now, to work out the solution,

Part (a)

First, we note that if the wheel is fair, each of five colors has equal chance of getting arrow ending in that color. This means that probability, p, that arrow ends in any particular color is 1/5 = 0.2.

Thus, our null hypothesis is: p = 0.2 ............................................................................................................ (3)

Given the rejection criterion that it takes three or fewer spins for a particular color to come up two times.

This essentially means that it takes two or three spins for a particular color to come up two times.

For any particular color to repeat twice in just 2 spins, the probability = 0.2²=0.04 ................................................ (4a)

If it is three spins, it would imply that in the first two spins, the particular color came up just once and it came up in the third spin. Hence, the probability = 2 x 0.2 x 0.8 x 0.2 = 0.064 ........................................... (4b)

(4a) and (4b) => P(2 or 3 spins for a particular color) = 0.104. Since this particular color can be any one of the five colors, P(2 or 3 spins) = 5 x 0.104 = 0.510

Thus, significance level is 0.501 or 51%. Option C Answer 1

Part (b)

Vide (2b),

Power = P(null hypothesis is rejected when p = 0.52 for purple and 0.12 for each of the other four colors.

Following the above analysis,

P(2 or 3 spins for a purple color)

= 0.52² + 2 x 0.52 x 0.48 x 0.52

= 0.2704 + 0.259584

= 0.529984.................................................................................................................................................. (5a)

P(2 or 3 spins for each of the other four colors)

= 0.12² + 2 x 0.12 x 0.88 x 0.12

= 0.0144 + 0.025344

= 0.039744

P(2 or 3 spins for four colors)

= 4 x 0.039744

= 0.158976 .................................................................................................................................................. (5ab)

(5a) and (5b) =>

Power = 0.529984 + 0.158976

= 0.68896 Option D Answer 2

DONE