A mathematics teacher wanted to see the correlation between test scores and
homework. The homework grade (x) and test grade (y) are given in the accompanying
table. Write the linear regression equation that represents this set of data, rounding
all coefficients to the nearest hundredth. Using this equation, estimate the homework
grade, to the nearest integer, for a student with a test grade of 68.
Homework Grade (x) Test Grade (y)
X | Y
88 | 90
55 | 55
89 | 91
85 | 88
61 | 52
76 | 76
76 | 81
61 | 59
Answers
Answer:
The homework grade, to the nearest integer, for a student with a test grade of 68 is 69.
Step-by-step explanation:
The general form of the linear regression equation is:
\(y=a+bx\)
Here,
y = dependent variable = test grade
x = independent variable = homework grade
a = intercept
b = slope
Compute the value of a and b as follows:
\(a=\frac{\sum Y\cdot \sum X^{2}-\sum X\cdot\sum XY}{n\cdot \sum X^{2}-(\sum X)^{2}}\\\\=\frac{(592\times 44909)-(591\times45227)}{(8\times44909)-(591)^{2}}\\\\=-14.316\) \(b=\frac{n\cdot \sum XY-\sum X\cdot\sum Y}{n\cdot \sum X^{2}-(\sum X)^{2}}\\\\=\frac{(8\times 45227)-(591\592)}{(8\times44909)-(591)^{2}}\\\\=1.195\)
The linear regression equation that represents the set of data is:
\(y=-14.316+1.195x\)
Compute the value of x for y = 68 as follows:
\(y=-14.316+1.195x\)
\(68=-14.316+1.195x\\1.195x=68+14.316\\1.195x=82.316\\x=68.884\\x\approx 69\)
Thus, the homework grade, to the nearest integer, for a student with a test grade of 68 is 69.
The linear regression equation which models the relationship between homework grade and test grade and the predicted homework grade are :
y = 1.20x - 14.32 Predicted, x = 69Using technology, the linear model can be obtained using either excel or a linear regression calculator.
Using a linear regression calculator which gives the linear equation in the form :
y = bx + cy = 1.20x - 14.32
y = Test grade ; x = homework grade Slope, b = 1.20 ; intercept, c = - 14.32Using the model equation obtained :
Test grade, y = 68Homework grade, xy = 1.20x - 14.32
68 = 1.20x - 14.32
68 + 14.32 = 1.20x
82.32 = 1.20x
x = (82.32 ÷ 1.20)
x = 68.6
x = 69 (nearest integer)
Therefore, the estimated homework grade for a test score of 68 is 69
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Step-by-step explanation:
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