Suppose a possibly biased die is rolled 30 times and that the face containing

two pips comes up 10 times. Do we have evidence to conclude that the die is biased?

p < α

0.01297 < 0.05

Since the p value is less than the α value therefore, we reject the null hypothesis so we have evidence to conclude that the die is biased.

Explanation:

H₀: The die is not biased

Ha: The die is biased

We can apply binomial distribution and determine whether the die is biased or not. (we can also perform z-test, it will provide similar results)

We know that a binomial distribution is given by

P (x; n, p) = nCx pˣ (1 - p) ⁿ⁻ˣ

Where p is the probability of success and 1 - p is the probability of failure, n is number of trials and x is the variable of interest.

For the given problem,

Total trials are n = 30

When you roll a die, there are total 6 possible outcomes,

The probability of getting the face containing two pips on each trial is

p = 1/6

p = 0.1667

The variable of interest is x = 10

P (10; 30, 0.1667) = ³⁰C₁₀*0.1667¹⁰ * (1 - 0.1667) ³⁰⁻¹⁰

P (10; 30, 0.1667) = (30045015) * (0.1667) ¹⁰ * (0.8333) ²⁰

P (10; 30, 0.1667) = 0.01297

Assuming that the level of significance is α = 0.05 then

p < α

0.01297 < 0.05

Since the p value is less than the α value therefore, we reject the null hypothesis so we have evidence to conclude that the die is biased.