Question

You play roulette and observe the ball land in the 1st dozen (i.e., 1-12) 5 times out of 5 spins.

You want to perform a test to determine if the wheel is fair.

a.

State the null and alternative hypothesis.

b.

What is the p-value of this test?

c.

For an∝=.05 significance level, what do you conclude?

Answer 1

Solution:

Part a

Here, we have to use z test for population proportion. The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: The roulette wheel is fair.

Alternative hypothesis: H_a: The roulette wheel is not fair.

H₀: p = 0.3333 versus H_a: p ≠ 0.3333

[There are total three dozens of numbers. So, probability that the ball land in first dozen is 1/3 = 0.3333.]

Part b

We are given n = 5 spins and the number of times a ball in first dozen = x = 5

Test statistic formula for one sample z test for population proportion is given as below:

Z = (P - p)/sqrt(p*(1 - p)/n)

Where, P is the sample proportion, p is the population proportion, Z is critical value, and n is sample size.

P = x/n = 5/5 = 1

Z = (1 – 0.3333) / sqrt(0.3333*(1 – 0.3333)/5)

Z = 3.1625

P-value = 0.0016

(by using z-table)

Part c

We are given

Level of significance = α = 0.05

P-value = 0.0016 < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that the roulette wheel is not fair.

There is insufficient evidence to conclude that the roulette wheel is fair.