Question

in a mathematics class of ten students, the teacher wanted to determine how a homework grade influenced a student’s performance on the subsequent test. the homework grade and subsequent test grade for each student are given in the accompanying table.
homework grade (x) | test grade (y)
94 | 98
95 | 94
92 | 95
87 | 89
82 | 85
80 | 78
75 | 73
65 | 67
50 | 45
20 | 40
what is the slope of this equation? round your answer to the nearest hundredth.

Explanation:

Step1: Calculate necessary sums

First, we need to calculate the sum of \( x \) (\( \sum x \)), sum of \( y \) (\( \sum y \)), sum of \( xy \) (\( \sum xy \)), and sum of \( x^2 \) (\( \sum x^2 \)) for the given data.

The data points are:
\( (x,y) \): (94,98), (95,94), (92,95), (87,89), (82,85), (80,78), (75,73), (65,67), (50,45), (20,40)

Calculating \( \sum x \):
\( 94 + 95 + 92 + 87 + 82 + 80 + 75 + 65 + 50 + 20 = 740 \)

Calculating \( \sum y \):
\( 98 + 94 + 95 + 89 + 85 + 78 + 73 + 67 + 45 + 40 = 764 \)

Calculating \( \sum xy \):
\( (94\times98)+(95\times94)+(92\times95)+(87\times89)+(82\times85)+(80\times78)+(75\times73)+(65\times67)+(50\times45)+(20\times40) \)
\( = 9212+8930+8740+7743+6970+6240+5475+4355+2250+800 = 60615 \)

Calculating \( \sum x^2 \):
\( 94^2 + 95^2 + 92^2 + 87^2 + 82^2 + 80^2 + 75^2 + 65^2 + 50^2 + 20^2 \)
\( = 8836+9025+8464+7569+6724+6400+5625+4225+2500+400 = 60768 \)

Step2: Use the slope formula for linear regression

The formula for the slope \( m \) of the linear regression line (which is appropriate here to find the relationship between homework grade \( x \) and test grade \( y \)) is:

\( m=\frac{n\sum xy - \sum x\sum y}{n\sum x^2 - (\sum x)^2} \)

where \( n = 10 \) (number of data points).

Substitute the values:

\( n = 10 \), \( \sum xy = 60615 \), \( \sum x = 740 \), \( \sum y = 764 \), \( \sum x^2 = 60768 \)

First, calculate the numerator:

\( n\sum xy - \sum x\sum y = 10\times60615 - 740\times764 \)
\( = 606150 - 565360 = 40790 \)

Then, calculate the denominator:

\( n\sum x^2 - (\sum x)^2 = 10\times60768 - 740^2 \)
\( = 607680 - 547600 = 60080 \)

Now, calculate the slope:

\( m=\frac{40790}{60080}\approx0.68 \) (rounded to the nearest hundredth)

Answer:

\( 0.68 \)