Question

Simon has a six-sided die that he suspects lands on the number 6 more frequently than would be predicted by chance alone. He amuses himself one day by repeatedly rolling the die and recording whether the outcome is a 6 or not. Out of 150 rolls, a 6 occurs 28 times. Simon decides to test whether the die is unfair, i.e., whether 6 does indeed occur more frequently than would be predicted by chance alone. He chooses a significance level of 5%.

a. What are the null and alternative hypotheses?

b. What is the value of the test statistic?

c. What is the p-value? Give an expression involving a probability, not just a final answer.

d. State your conclusions in the language of the problem.

e. Give a 95% confidence interval for the probability of rolling a 6 using Simon’s die.

Answer 1

Here the claim is "Die is unfair" that is p is not equal to 1/6

a. Null and alternative hypothesis:

b. Test statistics:

here p = population proportion which is 1/6

where x = number of times 6 occurs that is 28 and

n = total number of times die roll which is 150.

Z = 0.66

c. P-value

The alternative sign contains not equal to sign so this is the two-tailed test. And the test statistics is positive.

So the formula of P- value for two-tailed test statistics and when the test statistics is positive is:

By using z table the probability P(Z > 0.66) is 0.2546,

P-value = 2 * 0.2546 = 0.5092

d. Conclusion:

Decision rule: If P value > alpha(level of significance) then fail to reject the null hypothesis otherwise reject the null hypothesis.

Here alpha = significance level = 5% = 0.05

So P value(0.5092) > alpha(0.05), so fail to reject the null hypothesis.

Conclusion: There is no sufficient evidence to support the claim that the die is unfair.

e. 95% confidence interval for population proportion:

The formula of the confidence interval is:

First, have to find Z critical value for 95% confidence level

c = 0.95

alpha = 1-0.95 = 0.05

alpha/2 = 0.025, 1 - (alpha/2) = 0.975

By using z table the critical value for area 0.975 is 1.96

Lower limit = 0.124311

Upper limit = 0.249023

That is the 95% confidence interval for the probability of rolling a 5 using Simon's die is

(0.1243, 0.2490)