A logistic regression model describes how the probability of voting for the Repu

Question

A logistic regression model describes how the probability of voting for the Republican candidate in a presidential election depends on the voter's total family income (, in thousands of dollars) in the previous year. The estimated equation for a particular sample of voters is 1.00+0.02r where plx)is the probability of a vote for the Republican candidate. (a) Find the estimated probability that a voter with income $10,000 will vote for the Republican candidate (b) Find the estimated probability that a voter with income $100,000 will vote for the Republican (c) Describe how the probability that a voter will vote for the Republican candidate depends on the d) Compare the odds of support for the Republican candidate between two voters that are $50 000 (e) At what income is the estimated probability of a vote for the Republican candidate equal to 08? candidate. voter's income. What coefficient in the logistic equation describes this relationship? apart in income.

Explanation / Answer

Ans a) Given income of the voter 10000 dollars which is x * 1000 so putting it in equation

ln(p(x)/(1-p(x)) = -1.00 + 0.02 * 10

                          = -1 + 0.2 = -0.8

p(x)/(1-p(x)) = e^(-0.8)

Solving this equation we get p(x) = 0.31 approximately

Ans b) So if income is 100 thousand dollars we will get

ln(p(x)/(1-p(x)) = -1.00 + 0.02 * 100

                          = -1 + 2 = 1

p(x)/(1-p(x)) = e^(1)

Solving this we will get p(x) = 0.7311 approximately

Ans c The probability of voter voting Republic will increase with the income increasing and the coeffecient of x that is 0.02 determines this and we can infer that the probability is directly proportional to the income because the coeffecient of x is positive that is 0.02.

Ans d Let the two probability be denoted by p1 and p2 then

p1 = e^(-1 + 0.02x)/(1 + e^(-1 + 0.02x))

and , p2 = e^(-1 + 0.02(x + 50))/(1 + e^(-1 + 0.02(x + 50))) = e^(0.02x)/(1 + e(0.02x))

So odds of support for Voter 1 = 1 / (1 + e^(-1 + 0.02x)) and for Voter 2 = 1 / (1 + e(0.02x))

So we can thus compare it.

Ans e Given p(x) = 0.8

We get -1 + 0.02x = ln(4)

Therefore, x = 119.315 approximately

So the income should be 119.315 thousand dollars approximately to have a probabilty of 0.8 that the user is going to vote for Republicans

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