
Concept explainers
To determine:
a) We can see from the table that with the increase in value of x
, there is decrease in value of y
, hence we can say that coefficient of
b) Equation of least squares is y=−2.9866x+24.92
and
Plot is

c) Predicted value of y at x=2.4 is 17.752
Given information: Five points
x | −4 | −3 | −1 | 3 | 5 |
y | 3.7 | 33.7 | 27.5 | 16.4 | 9.8 |
Formula used: Slope, m=n∑i=1(xi−ˉX)(yi−ˉY)n∑i=1(xi−ˉX)2
Y axis intercept, b=ˉY−mˉX
Correlation Coefficient,r=n∑i=1(xi−ˉX)(yi−ˉY)√n∑i=1(xi−ˉX)2√n∑i=1(yi−ˉY)2
Where xi and yi are the i th entry of x and y ; ˉX and ˉY are means of x and y
Calculation:
Step 1: Calculate the mean of x and y
ˉX=(−4)+(−3)+(−1)+3+55=0ˉY=37.2+33.7+27.5+16.4+9.85=24.92Step 2: Plot the table as shown
i | xi | yi | xi−ˉX | yi−ˉY | (xi−ˉX)2 | (xi−ˉX)(yi−ˉY) | (yi−ˉY)2 |
1 | -4 | 37.2 | -4 | 12.28 | 16 | -49.12 | 150.7984 |
2 | -3 | 33.7 | -3 | 8.78 | 9 | -26.34 | 77.0884 |
3 | -1 | 27.5 | -1 | 2.58 | 1 | -2.58 | 6.6564 |
4 | 3 | 16.4 | 3 | -8.52 | 9 | -25.56 | 72.5904 |
5 | 5 | 9.8 | 5 | -15.12 | 25 | -75.6 | 228.6144 |
n∑i=1(xi−ˉX)2=60 | n∑i=1(xi−ˉX)(yi−ˉY)=−179.2 | n∑i=1(yi−ˉY)2=535.7480 |
Step 3: Calculate the slope m=n∑i=1(xi−ˉX)(yi−ˉY)n∑i=1(xi−ˉX)2=−179.260≈−2.9866
Step 4: Calculate y intercept b=ˉY−mˉX=24.92−(−2.9866×0)≈24.92
The slope of the line is −2.9866 and y intercept is 24.92
Using slope intercept form, y=mx+b ,equation is y=−2.9866x+24.92;
Step 5: Draw the

Step 6: Calculate the Correlation coefficient
r=n∑i=1(xi−ˉX)(yi−ˉY)√n∑i=1(xi−ˉX)2√n∑i=1(yi−ˉY)2=−179.2√60×√535.7480≈−0.9995Step 7: Prediction for x=2.4 ; plug x=2.4 in y=−2.9866x+24.92
y=−2.9866(2.4)+24.92=17.752
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Chapter 2 Solutions
College Algebra with Modeling & Visualization (5th Edition)
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