chapter 12.3 regression equation

linear regression

The linear relationship if strong if the points are close to a straight line, except in the case
of a horizontal line where there is no relationship.
If we think that the points show a linear relationship, we would like to draw a line on the
scatter plot.

line of best fit

a line on a graph showing the general direction
that a group of points seem to follow.

the line

Our aim is to calculate the values 𝒃 (slope) and 𝒂 (y-intercept) in the equation of a line:
𝑦 = 𝑎 + 𝑏𝑥
Where:
• 𝑦 = how far up
• 𝑥 = how far along
• 𝑏 = slope (how steep the line is)
• 𝑎 = 𝑦-intercept (where the lines crosses the 𝑦 axis)

To find the line of best fit for n points

Step 1: For each (𝑥, 𝑦) point, calculate 𝑥2 and 𝑥𝑦
Step 2: Sum all 𝑥, 𝑦, 𝑥2, and 𝑥𝑦, which gives us σ 𝑥, σ 𝑦, σ 𝑥2, and σ 𝑥𝑦 (σ means “sum up”)
Step 3: Calculate slope 𝒃:
𝑏 = 𝑛 σ 𝑥𝑦 − σ 𝑥 σ 𝑦
𝑛 σ 𝑥2 − σ 𝑥 2
(where 𝑛 is the number of points)
Step 4: Calculate Intercept 𝒂:
𝑎 = σ 𝑦 − 𝑚 σ 𝑥
𝑛
Step 5: Assemble the equation of a line
𝑦 = 𝑎 + 𝑏𝑥

UNDERSTANDING SLOPE

slope of the line, 𝑏, describes how changes in the variables are related

INTERPRETATION OF THE SLOPE

The slope of the best-fit line tells us how the
dependent variable (𝑦) changes for every one unit increase in the independent (𝑥)
variable, on average.

CALCULATOR GUIDE using the Linear Regression T Test: LinRegTTes

1. In the STAT list editor, enter the 𝑋 data in list L1 and the 𝑌 data in list L2, paired so
that the correspond (𝑥, 𝑦) values are next to each other in the lists.
2. On the STAT TESTS menu, scroll down with the cursor to select the LinRegTTest.
3. On the LinRegTTest input screen enter: Xlist: L1 ; Ylist: L2; Freq: 1
4. On the next line, at the prompt 𝛽 or 𝜌, highlight “≠ 0” and press ENTER
5. Leave the line for “RegEq:” blank
6. Highlight Calculate and press ENTER

focus on only some information,
specifically: 𝑎, 𝑏, 𝑟, and 𝑟2

CALCULATOR GUIDE
Graphing the Scatterplot and Regression Line

1. We are assuming your 𝑋 data is already entered in list L1 and your Y data is in list L2
2. Press 2nd STATPLOT ENTER to use Plot 1
3. On the input screen for Plot 1, highlight On, and press ENTER
4. For TYPE: highlight the very first icon which is the scatterplot and press ENTER
5. Indicate Xlist: L1 and Ylist: L2
6. For Mark: it does not matter which symbol you highlight.
7. Press the ZOOM key and then the number 9 (for menu item “ZoomStat”); the
calculator will fit the window to the data
8. To graph the best-fit line, press the “Y=“ key and type the equation −173.5 + 4.83𝑋
into equation Y1. (The X key is immediately left of the STAT key). Press ZOOM 9 again
to graph it.
9. Optional: If you want to change the viewing window, press the WINDOW key. Enter
your desired window using Xmin, Xmax, Ymin, Ymax

CORRELATION COEFFICIENT

𝒓, as a measure of strength and direction of the
linear relationship between 𝑥 and 𝑦 .

correlation coefficient

What the VALUE of 𝒓 tells us:
• It is always between −1 and 1: −1 ≤ 𝑟 ≤ 1
• The size of the correlation 𝑟 indicates the strength of the linear relationship between 𝑥
and 𝑦.
• Values of 𝑟 close to −1 or 1 indicate a stronger linear relationship.
If 𝑟 = 0 there is likely no linear correlation.
If 𝑟 = 1, there is a perfect positive correlation.
If 𝑟 = −1, there is perfect negative correlation.

What the SIGN of 𝒓 tells us

positive value of 𝑟 means that when 𝑥 increases, 𝑦 tends to increase and when 𝑥
decreases, 𝑦 tends to decrease (positive correlation).
• A negative value of 𝑟 means that when 𝑥 increases, 𝑦 tends to decrease and when 𝑥
decreases 𝑦 tends to increase (negative correlation).
• The sign of 𝑟 is the same as the sing of the slope 𝑏 of the best-fit line.

warning

Strong correlation does not suggest that 𝑥 causes 𝑦 or 𝑦 causes 𝑥.
We say “correlation does not imply causation