Conduct the hypothesis test and provide the test statistic, critical value and P

Question

Conduct the hypothesis test and provide the test statistic, critical value and P-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: 27, 32, 41, 41, 26, 33. 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.The critical value is.The P-value is.state the conclusion. H-0.There sufficient evidence to support the claim that the outcomes are not equally likely. The outcomes to be equally likely, to behave differently from a fair die.

Explanation / Answer

Here we have to test the hypothesis that,

H0 : Outcomes are equally likely.

H1 : Outcomes are not equally likely.

Assume alpha = level of significance = 0.01

We are given that outcomes and their observed frequencies.

We know that distribution of tossing a single die is Uniform.

So probability of each putcome is same.

P(1) = P(1) = P(3) = P(4) = P(5) = P(6) = 1/6 = 0.1667

These are the probabilities of each outcome.

So the complete table of x and p is,

And the test statistic is,

X2 =   (O - E)2 / E

where O is observed frequency and

E is expected frequency.

These probabilities we apply for calculating expected frequency.

The formula of expected frequency is,

E = N*p

where N = number of times die roll.

The table of x, observed frequency and expected frequency is,

The test statistic is,

X2 = 6.4

P-value we can find by using EXCEL.

syntax is,

=CHIDIST(x, deg_freedom)

where x is test statistic value.

deg_freedom = n-1 = 6 - 1 = 5

P-value = 0.2692

Critical value also we can find by using EXCEL.

syntax is,

=CHIINV(probability, deg_freedom)

where probability = alpha/2

deg_freedom = n-1

critical value = 16.7496

X2 < critical value or P-value > alpha

Accept H0 at 1% level of significance.

Conclusion : : There is sufficient evidence to say that outcomes are equally likely.

x p 1 0.166667 2 0.166667 3 0.166667 4 0.166667 5 0.166667 6 0.166667 total 1

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