Question

The following bivariate data set contains an outlier.

x	y
47.6	53.7
40.6	112.6
28.9	72.2
36.1	101.1
32.4	112.4
26.8	67.1
36.8	38.9
33.1	70.1
67.5	8.6
43.3	-6.4
50.1	41.3
30.1	3.2
29.8	41.7
55	-32.8
189.6	548.8

What is the correlation coefficient with the outlier?
r_w =

What is the correlation coefficient without the outlier?
r_wo =

Would inclusion of the outlier change the evidence for or against a significant linear correlation at 5% significance?

A) No. Including the outlier does not change the evidence regarding a linear correlation.

B) Yes. Including the outlier changes the evidence regarding a linear correlation.

Would you always draw the same conclusion with the addition of an outlier?

A) Yes, any outlier would result in the same conclusion.

B) No, a different outlier in a different problem could lead to a different conclusion.

Explain your answer.

Answer 1

a)

using excel data analysis tool for regression, following o/p is obtained

Regression Statistics
Multiple R	0.8680
R Square	0.7534
Adjusted R Square	0.7344
Standard Error	70.1143
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	1	195208.35	195208.35	39.7087	0.0000
Residual	13	63908.14	4916.01
Total	14	259116.49
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-63.9811	29.4216	-2.1746	0.0487	-127.543	-0.420
X	2.9319	0.4653	6.3015	0.0000	1.927	3.937

so, correlation coefficient with the outlier=0.8680

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correlation coefficient without the outlier= -0.4853
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with outlier,

correlation hypothesis test      
Ho:   ρ = 0  
Ha:   ρ ╪ 0  
n=   15  
alpha,α =    0.1  
correlation , r=   0.8680  
t-test statistic =    t = r*√(n-2)/√(1-r²) =    6.3015
critical t-value =    1.7709  
p-value =    0.0000 <α=0.05, reject Ho, so, linear correlation exists at α=0.05

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now without lier

correlation hypothesis test      
Ho:   ρ = 0  
Ha:   ρ ╪ 0  
n=   14  
alpha,α =    0.1  
correlation , r=   -0.4853  
t-test statistic =    t = r*√(n-2)/√(1-r²) =    -1.9227
critical t-value =    1.7823  
p-value =    0.0786 >α=0.05, fail to reject Ho, so, linear correlation does not exists at α=0.05

hence, answer is Yes. Including the outlier changes the evidence regarding a linear correlation.

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No, a different outlier in a different problem could lead to a different conclusion.