Question

The graph below shows the relationship between the number of months different students practiced baseball and the number of games they won:

The title of the graph is Baseball Games. On x axis, the label is Number of Months of Practice. On y axis, the label is Number of Games Won. The scale on the y axis is from 0 to 22 at increments of 2, and the scale on the x axis is from 0 to 12 at increments of 2. The points plotted on the graph are the ordered pairs 0, 1 and 1, 3 and 2, 5 and 3, 9 and 4, 10 and 5, 12 and 6, 13 and 7, 14 and 8,17 and 9, 18 and 10,20. A straight line is drawn joining the ordered pairs 0, 1.8 and 2, 5.6 and 4, 9.2 and 6, 13 and 8, 16.5 and 10, 20.5.

Part A: What is the approximate y-intercept of the line of best fit and what does it represent? (5 points)

Part B: Write the equation for the line of best fit in slope-intercept form and use it to predict the number of games that could be won after 13 months of practice. Show your work and include the points used to calculate the slope. (5 points)

Answer 1

(A) The y-intercept of the line is 1.82.

(B) The number of games that could be won after 13 months of practice is 26.

Explanation:

The data from the provided graph is:

X Y

0 1

1 3

2 5

3 9

4 10

5 12

6 13

7 14

8 17

9 18

10 20

Here,

X : Number of Months of Practice

Y : Number of Games Won

(A)

Compute the y-intercept of the line as follows:

`a=(\sum Y\cdot \sum X^(2)-\sum X\cdot \sum XY)/(n\cdot \sum X^(2)-(\sum X)^(2))`

`=(122\cdot 385-55\cdot 814)/(11\cdot 385-(55)^(2))\\\\=1.818\\\\\approx 1.82`

The y-intercept of the line is 1.82.

The y-intercept is the average value of the dependent variable, here the number of games won, when the value of the independent variable, here number of months of practice, is 0.

So, a y-intercept of 1.82 indicates that on average 1.82 can be won if the number of months of practice is 0.

(B)

Compute the slope as follows:

`b=(n\cdot \sum XY-\sum X\cdot \sum Y)/(n\cdot \sum X^(2)-(\sum X)^(2))`

`=(11\cdot 814-55\cdot 122)/(11\cdot 385-(55)^(2))\\\\=1.855\\\\\approx 1.86`

The equation for the line of best fit in slope-intercept form is:

`y=1.82+1.85x`

Predict the number of games that could be won after 13 months of practice as follows:

`y=1.82+1.85x`

`=1.82+(1.85\timex 13)\\=25.87\\\approx 26`

Thus, the number of games that could be won after 13 months of practice is 26.