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,..., Ps and that n300 responses produced these category counts. Category 1 23 4 5 Observed Count 48 62 76 51 63 (a) Are the five categories equally likely to occur? How would you test this hypothesis? Ho: At least one pi is different from Hai p1 = p2 = p3 = p4 = p5 = Ha: At least one pi is different from Ho: p1 = p2 = p3 = p4 = p5 = 0 Ha: At least one pi is different from 0 Ho: p1 = p2 = p3 = p4 = p5 = 1 Ha: At least one p, is different from1 Ho: At least one pi is different from 0 Ha: p1 = p2 = p3 = p4 = p5 = 0 (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.) 05 (d) Calculate the observed value of the test statistic. (Round your answer to two decimal places.) (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 insufficient evidence to indicate that at least one category is more likely to occur than the others

Explanation / Answer

Ans:

a)

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

Ha:Atleast one pi is different from 1/5

b)number of categories=n=5

Degree o freedom=5-1=4

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

d)

Test statistic:

calculated chi square score=8.23

e)

As,test stastic=8.23<9.488,we fail to reject null hypothesis.

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

(second option is correct)

Observed(O) Expected(E) (O-E)^2/E 1 48 60 2.4 2 62 60 0.067 3 76 60 4.267 4 51 60 1.35 5 63 60 0.15 Total 300 300 8.233

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