Question

Conduct the hypothesis test and provide the test statistic and the critical value, and state the conclusion.

A person drilled a hole in a die and filled it with a lead weight, then proceeded to roll it 200 times. Here are the observed frequencies for the outcomes of 1, 2, 3, 4, 5, and 6, respectively: 28, 27, 40, 38, 26, 41. Use a 0.01 significance level to test the claim that the outcomes are not equally likely. Does it appear that the loaded die behaves differently than a fair die?

The test statistic is __________.

(Round to three decimal places as needed.)

The critical value is __________ .

(Round to three decimal places as needed.)

State the conclusion.

(1) __________ Ho . There (2) __________ sufficient evidence to support the claim that the outcomes are not equally likely. The outcomes (3) __________ to be equally likely, so the loaded die (4) __________ to behave differently from a fair die.

(1) Do not reject

Reject

(2) is

is not

(3) appear

do not appear

(4) does not appear

appears

Answer 1

Answer: A person drilled a hole in a die and filled it with a lead weight, then proceeded to roll it 200 times. Here are the observed frequencies for the outcomes of 1, 2, 3, 4, 5, and 6, respectively: 28, 27, 40, 38, 26, 41. Use a 0.01 significance level to test the claim that the outcomes are not equally likely. Does it appear that the loaded die behaves differently than a fair die?

Solution:

The hypothesis test:

Ho: Outcomes are equally likely.

Ha: Outcomes aren't equally likely.

Test statistic χ2:

χ2 = (Observed value - Expected value)^2/Expected value

Outcomes	Observed (O)	Expected (E)	(O-E)^2/E
1	28	200/6 = 33.3333	0.8533
2	27	33.3333	1.2033
3	40	33.3333	1.3333
4	38	33.3333	0.6533
5	26	33.3333	1.6133
6	41	33.3333	1.7634
Total	200	200	7.4200

Therefore,

Test statistic χ2 = 7.420

Critical value of χ2:

df = k-1 = 6-1 = 5

Critical value of χ2(df, α) = 15.086

Since, test statistic χ2 (7.420) < critical value (15.086)

We fail to reject null hypothesis Ho.

Conclusion:

Do not reject Ho . There is not sufficient evidence to support the claim that the outcomes are not equally likely. The outcomes appears to be equally likely, so the loaded die does not appear to behave differently from a fair die.

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