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

and that n = 300 responses produced these category counts.

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

H0: p1 = p2 = p3 = p4 = p5 = 1

Ha: At least one pi is different from 1.

H0: p1 = p2 = p3 = p4 = p5 = 0

Ha: At least one pi is different from 0.

    H0: At least one pi is different from

.
Ha: p1 = p2 = p3 = p4 = p5 =

H0: At least one pi is different from 0.

Ha: p1 = p2 = p3 = p4 = p5 = 0

H0: p1 = p2 = p3 = p4 = p5 =

Ha: At least one pi is different from

.
(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.)

?20.05 =

(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

Category 1 2 3 4 5 Observed Count   49 61 76 51 63

Explanation / Answer

Ans:

a)

H0: p1 = p2 = p3 = p4 = p5 =1/5

Ha: At least one pi is different from 1/5

b)k=5

df=5-1=4

c)critical chi square value=CHIINV(0.05,4)=9.488

d)

Observed value of test statistic=7.80

e) As, test statistic is less than 9.488,we fail to reject the null hypothesis.

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

Observed(fo) Expected(fe) (fo-fe)^2/fe 1 49 60 2.02 2 61 60 0.02 3 76 60 4.27 4 51 60 1.35 5 63 60 0.15 Total= 300 300 7.80

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