4. Are couch potatoes (people who watch a lot of television) experts in sports?

Question

4. Are couch potatoes (people who watch a lot of television) experts in sports? A survey was conducted to determine whether a person is an expert in a sport merely by watching a lot of it on television. Thus, the question is: Does watching a lot of sports on television mean that one is an expert in that/those sport(s)? Here is a table of the gathered information. To qualify as an expert in sports, a sports knowledge exam was given to all participants. Couch potato Not a couch potato46 Not an expert in sports 37 61 56 The hypotheses are as follows: Ho: Sports expertise and watching a lot a television are independent variables Ha: Sports expertise and watching a lot a television are dependent variables Conduct the test for independence at the alpha = 0.05 level of significance. Calculate the test statistic and the p-value. Find the Expected Values for each cell in the table. Make a decision. a. b. Conduct a two-proportion z-test, with alpha = 0.05. Determine the critical region for the hypotheses Ho: .-, vs. Ha_ , (the proportion of couch potatoes who are experts in sports is the same as the proportion of non-couch potatoes who are experts in sports) Calculate the test statistic and p-value for this test. Make a decision.

Explanation / Answer

(a)

here we used chi-square test and chi-square=sum(O-E)2/E=5.91 with (r-1)(c-1)=(2-1)*(2-1)=1

p-value=0.0151 ( using ms-excel=chidist(5.91,1)

critical chi-square(0.05,1)=3.84 is less than calculated chi-square=5.91, so we fail to accept H0.

(b) p1=56/93=0.60, p2=46/107=0.43

var(p1-p2)=var(p1)+var(p2)=p1(1-p1)/n1 + p2(1-p2)/n2=0.6*(1-0.6)/93 + 0.43*(1-0.43)/107=0.004871

SE(p1-p2)=sqrt(var(p1-p2)=sqrt(0.004871)=0.0698

here z-test will be used and z=(p1-p2)/(SE(p1-p2)=(0.6-0.43)/0.0698=2.44

p-value=0.0074 is less than alpha=0.05 so we reject (or fail to accept) H0

two tailed critical z(0.05)=1.96 is less than calculated z=2.44, so we fail to accept H0

observed (O) Expected Expected(E) (O-E) (O-E)2/E 56 93*102/200= 47.4 8.6 1.55 46 107*102/200= 54.6 -8.6 1.35 37 93*98/200= 45.6 -8.6 1.61 61 107*98/200= 52.4 8.6 1.40 sum= 200 0 200 0 5.91

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