Hey, can someone help me with this problem please A survey of voter sentiment wa
ID: 3170757 • Letter: H
Question
Hey, can someone help me with this problem please
A survey of voter sentiment was conducted in four midcity political wards to compare the fraction of voters favoring candidate A. Random samples of 200 voters were polled in each of the four wards, with the results as shown in the accompanying table. The numbers of voters favoring A in the four samples can be regarded as four independent binomial random variables. Construct a likelihood ratio test of the hypothesis that the fractions of voters favoring candidate A are the same in all four wards. Use alpha = .05.Explanation / Answer
Solution:
The log likelihood is -2ln(null hypothesis) +2ln(likelihood for all alterntive model)
As the combinatorials divide out, we have
-2 (236 ln(236/800)+ 564 ln(564/100) )
+ 2 (76 ln(76/200)+124 ln(124/200))
+ 2 (53 ln(53/200)+147 ln(147/200))
+ 2 (59 ln(59/200)+141 ln(141/200))
+ 2 (48 ln(48/200)+152 ln(152/200))
= 10.53527.
Under the null hypothesis, this should be distributed as the Chi-squared statistic with 4-1 = 3 degrees of freedom.
alpha = .05
Excel has a function, chidist(x,degrees of freedom) which will give the tail probability, or the p-value
Then, chidist(10.53527,3) = .014524. This is the p-value
The critical value for alpha = .05 is chiinv(.05,3) = 7.814728
10.53527 > 7.814728, so reject the null hypothesis. Alternatively, the p-value .014524 < alpha = .05, so reject the null hypothesis.