Question

YEAR	DISTRICT	DMAC	RMAC	DABS	RABS
1982	2	47767	21340	551	205
1982	4	44437	28533	594	312
1982	8	55662	13214	338	115
1984	1	58327	38883	1357	764
1984	3	78270	6473	716	144
1984	5	54812	55829	1207	1436
1984	7	77136	13730	929	258
1986	2	39034	23363	609	316
1986	4	52817	16541	666	306
1986	8	48315	11605	477	171
1988	1	56362	34514	1101	700
1988	3	69801	3939	448	70
1988	5	43527	56721	781	1610
1988	7	68702	12602	644	250
1990	2	27543	26843	660	509
1990	4	39193	27664	482	831
1990	8	34598	8551	308	148
1992	1	65943	21518	1923	594
1992	3	58480	12968	695	327
1992	5	41267	46967	841	1275
1992	7	65516	14310	814	423
1993	2	19127	19691	1396	371

We need to examine the possible discrepancy in the absentee votes and compare it to a corresponding measure for the machine votes. Use Data-Compute- Expression to create a new variable for the difference between the democratic absentee vote count and the republican absentee vote count. Name this DIFFABS for absentee difference between the two parties. Create another new variable for the difference between the democratic machine vote and the republican machine vote. Name this variable DIFFMAC.

Find a regression line to predict the difference in absentee votes, DIFFABS, from the difference in machine votes, DIFFMAC.

What is the slope of this regression line? Interpret.

What is the intercept for this regression line? Interpret this intercept.

Make a prediction: If the machine vote count difference was 10,000 votes what would you predict for the absentee vote count difference?

What is the correlation coefficient (r) between the absentee vote count difference and the machine vote count difference?

What is the r² value and how do you interpret it?

Plot the residuals vs. the x-values. Is there a pattern?

Save the residuals.

Make a boxplot of the residuals to identify any outliers.

Which years:districts have residuals that are outliers in this fit? Is the 1993 district 2 election one of them?

Answer 1

Solution

Let x = DIFFMAC = DMAC – RMAC and y = DIFFABS = DABS - RABS

Back-up Theory

The linear regression model Y = β₀ + β₁X + ε, ……………………………………..(1)

where ε is the error term, which is assumed to be Normally distributed with mean 0 and variance σ².

Estimated Regression of Y on X is given by: Y = β_0cap + β_1capX, ……………….(2)

where β_1cap = Sxy/Sxx and β_0cap = Ybar – β_1cap.Xbar..…………………..….…..(3)

Mean X = Xbar = (1/n)sum of x_i ………………………………………….……….….(4)

Mean Y = Ybar = (1/n)sum of y_i ………………………………………….……….….(5)

Sxx = sum of (x_i – Xbar)² ………………………………………………..…………....(6)

Syy = sum of (y_i – Ybar)² ……………………………………………..………………(7)

Sxy = sum of {(x_i – Xbar)(y_i – Ybar)} ……………………………………………….(8)

All above sums are over i = 1, 2, …., n,n = sample size ………………………...(9)

Correlation coefficient, r = Sxy/√{Sxx . Syy} …………………………………… (10)

Now to work out the solution,

Summary of Excel Calculations:

n	22
Xbar	32706.60
ybar	341.7
Sxx	7330734333
Syy	3780764.42
Sxy	66140893.2
β1cap	0.00902241
β0cap	46.6076343
r	0.39728897
r^2	0.15783852

Answers

Regression line to predict the difference in absentee votes, DIFFABS, from the difference in machine votes, DIFFMAC: DIFFABS = 46.61 + 0.0090 DIFFMAC ANSWER 1

Slope of the regression line = 0.0090 ANSWER 2

Interpretation of slope:

For a difference of one unit in DIFFMAC, the increase in DIFFABS would be 0.0090. ANSWER 3

Intercept for the regression line = 46.61 ANSWER 4

Interpretation of intercept: Even when DIFFMAC is zero, DIFFABS would be 46.61. ANSWER 5

If the machine vote count difference was 10,000 votes, prediction for the absentee vote count difference would be: 46.61 + (10000 x 0.0090) = 136.61 ANSWER 6

Correlation coefficient (r) between the absentee vote count difference and the machine vote count difference = 0.3973 ANSWER 7

r² value = 0.1578 A =NSWER 8

Interpretation of r² value: r² value represents the proportion of variation in DIFFABS that is explained by DIFFMAC. ANSWER 9

DONE