Suppose that a response can fall into one of k = 5 categories with probabilities

Question

Suppose that a response can fall into one of k = 5 categories with probabilities p1, p2, , p5 and that n = 300 responses produced these category counts.

Category 1 2 3 4 5

Observed Count 46 63 73 52 66

(a) Are the five categories equally likely to occur? How would you test this hypothesis?

H0: At least one pi is different from 1 5 . Ha: p1 = p2 = p3 = p4 = p5 = 1 5

H0: p1 = p2 = p3 = p4 = p5 = 0 Ha: At least one pi is different from 0.

H0: p1 = p2 = p3 = p4 = p5 = 1 5 Ha: At least one pi is different from 1 5 .

H0: At least one pi is different from 0. Ha: p1 = p2 = p3 = p4 = p5 = 0

H0: p1 = p2 = p3 = p4 = p5 = 1 Ha: At least one pi is different from 1.

(b) If you were to test this hypothesis using the chi-square statistic, how many degrees of freedom would the test have?

______ degrees of freedom

(c) Find the critical value of ?2 that defines the rejection region with ? = 0.05. (Round your answer to three decimal places.)

?2 0.05 =

(d) Calculate the observed value of the test statistic.

?^2 =

(e) Conduct the test and state your conclusions.

There is sufficient evidence to indicate that at least one category is more likely to occur than the others.

There is sufficient evidence to indicate that there is not at least one category more likely to occur than the others.

There is insufficient evidence to indicate that at least one category is more likely to occur than the others.

There is insufficient evidence to indicate that there is not at least one category more likely to occur than the others.

You may need to use the appropriate appendix table or technology to answer this question.

Explanation / Answer

a) Correct answer: H0: p1 = p2 = p3 = p4 = p5 = 1 / 5 Ha: At least one pi is different from 1 / 5 .

b) 5-1 = 4 df

c) Critical X^2: 9.48772

d)

Test Statistic, X^2: 7.9000

e) Correct answer: There is sufficient evidence to indicate that there is not at least one category more likely to occur than the others.

Observed Expected Category Freq (Oi) Freq Ei (Oi-Ei)^2 /Ei 1 46 60 3.266666667 2 63 60 0.15 3 73 60 2.816666667 4 52 60 1.066666667 5 66 60 0.6 Total: 300 300 7.9

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