Given below are seven observations collected in a regression study on two variab
ID: 3223969 • Letter: G
Question
Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). Use Excel's Regression Tool to answer the following questions.
x
y
2
12
3
9
6
8
7
7
8
6
7
5
9
2
a.
What is the estimated regression equation?
b.
Perform a t-test and determine whether or not x and y are related. Use a = 0.05.
c.
Perform an F-test and determine whether or not x and y are related. Use a = 0.05.
d.
Find and interpret the coefficient of determination.
STEP BY STEP SOLUTION PLEASE!
x
y
2
12
3
9
6
8
7
7
8
6
7
5
9
2
Explanation / Answer
(a)
We will find an equation of the regression line in 4 steps.
Step 1: Find XY and X as it was done in the table below.
Step 2: Find the sum of every column:
X=42 , Y=49 , XY=249 , X2=292
Step 3: Use the following equations to find a and b:
a=YX2XXY/nX2(X)2=4929242249729242213.75
b=nXYXY/nX2(X)2=724942497292(42)21.125
Step 4: Substitute a and b in regression equation formula
y = a + bx
y = 13.75 1.125x
(b)
Null Hypothesis (Ho): X and Y are not related
Alternative Hypothesis (H1): X and Y are related
For the given data we get the below values
Observed difference (Sample 1 - Sample 2): -1
Standard Deviation of Difference : 1.543
Test statistic
t = (Mean (X) – Mean (Y))/Sqrt ((s12/n1)+ (s22/n2))
Unequal Variances
DF : 11
By using the above (t) formula we get
T-Value -0.6481
Population 1 Population 2: P-Value = 0.5302
Conclusion: The result is not significant at p<0.05,
Therefore accept Null Hypothesis (Ho)
i.e X and Y are not related
(C)
Null Hypothesis (Ho): X and Y are not related
Alternative Hypothesis (H1): X and Y are related
Conclusion: Fcaluculated value < Fcritical value
i.e 0.42 < 4.7472
Therefore we accept Null Hypothesis (Ho)
i.e X and Y are not related
(d)
Step 1: Find XY , X2 and Y2 as it was done in the table below.
Step 2: Find the sum of every column to get:
X=42 , Y=49 , XY=249 , X2=292 , Y2=403
Step 3: Use the following formula to work out the correlation coefficient.
r = (nXYXY)/sqrt[nX2(X)2][nY2(Y)2]
r = 0.9186
The coefficient of determination r2 = 0.8438
Interpretation:
Since r2 is between 0 and 1 " r2x100 percent of the variation in y is reduced by taking into account predictor X".
X Y XY XX 2 12 24 4 3 9 27 9 6 8 48 36 7 7 49 49 8 6 48 64 7 5 35 49 9 2 18 81Related Questions
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