Please provide complete rigorous solution! A simple ran candidates A and B. 50%)

Question

Please provide complete rigorous solution!

A simple ran candidates A and B. 50%) in favor of one candidate or the other. (We assume that all responses were eith dom sample of 200 students were surveyed and asked their preference between We wish to test whether or not there is a significant majority (more than (a) Use a normal approximation (20.025 1.96) to find a critical value for a level = 0.05 test if the test statistic is D = the number expressing support for A-the number expressing support for B (b) Calculate the power of this test against the alternative that 45% of the population sup- ports candidate A.

Explanation / Answer

Here if test statistic is D, which is

D = l number expressing support for A - number expressing support for B l

number expressing support for A + number expressing support for B = 200

so D can also be expressed as

D = l 2 * number expressing support for A - 200 l

Here hypothesiszed of D = 0

here pA = pB = 0.5 [ in the case of null hypothesis]

standard error of D se0 = sqrt [2 * 0.5 * 0.5 * 200] = 10

so The critical values are (as it is two tailed test) = Dbar +- Z0.05 se0   = 0 + 1.96 * 10 = +- 19.6

so the critical values are + 19.60. (as D takes only positi value)

that critical value means if the difference values are more than + 19.60 then we can reject the null hypothesis that support for both the candiates are not equal.

(b) Here Let say Support for candidate A = 0.45

so expected support for candidate A = 0.45 * 200 = 90

expected support for candiate B = 0.55 * 200 = 110

Expeced(D) = l 90 - 110 l = 20

standard deviation of support for the candidate A = sqrt (2 * 0.45 * 0.55 * 200) = 9.95

so now we want to calculate power of the test that means we want to know the probability tha we will correctly detect that the null hypothesis is incorrect.

Power = Pr(D > 20) = NORM( D > 20 ; 19.6 ; 9.95) = 1 - Norm (D < 20 ; 19.6 ; 9.95)

Z = (20 - 19.6)/ 9.95 = 0.04

Power = Pr(D > 20) = 1 - Pr( Z < 0.04) = 1 - 0.5160 = 0.4840

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