Example 1

Given the equation y = 2.3x + 5:

1. Create an x-y table to describe the values of at least four points
2. What is the slope of the line?
3. What is the y-intercept?

Solution

1. Pick a value for x, substitute the chosen value for x in the equation, and calculate y:

   Table 1
   x	calculation	y
   1	y = 2.3(1) + 5	7.3
   2	y = 2.3(2) + 5	9.6
   0	y = 2.3(0) + 5	5
   −1	y = 2.3(−1) + 5	2.7

   The equation in the problem is in y = mx + b form (also known as slope-intercept form), where b is the y-value when x = 0, and m is the slope of the line.
2. m = 2.3
3. b = 5

Example 2

Given the equation y = −3x + 3.9:

1. Create an x-y table to describe the values of at least four points
2. What is the slope of the line?
3. What is the y-intercept?

Solution

1. Pick a value for x, substitute the chosen value for x in the equation, and calculate y:

   Table 2
   x	calculation	y
   1	y = −3(1) + 3.9	6.9
   2	y = −3(2) + 3.9	9.9
   0	y = −3(0) + 3.9	3.9
   −1	y = −3(−1) + 3.9	0.9

   The equation in the problem is in y = mx + b form (also known as slope-intercept form), where b is the y-value when x = 0, and m is the slope of the line.
2. m = −3
3. b = 3.9

Example 3

Given the equation y = −2.8x − 9.1:

1. Create an x-y table to describe the values of at least four points.
2. What is the slope of the line?
3. What is the y-intercept?

Solution

1. Pick a value for x, substitute the chosen value for x in the equation, and calculate y:

   Table 3
   x	calculation	y
   1	y = −2.8(1) − 9.1	−11.9
   2	y = −2.8(2) − 9.1	−14.7
   0	y = −2.8(0) − 9.1	−9.1
   −1	y = −2.8(−1) − 9.1	6.3

   The equation in the problem is in y = mx + b form (also known as slope-intercept form), where b is the y-value when x = 0, and m is the slope of the line.
2. m = −2.8
3. b = −9.1

Guided Practice

1. If a linear graph exhibits a positive slope, what can you predict will happen to the response variable as the explanatory variable increases?
2. If a linear graph has no slope, what does that mean?
3. Given the linear equation 2y = 5.2x + 7:
   1. What is the slope?
   2. What is the y-interept?
   3. What happens to y as x increases?
4. Given the equation y = 2x² +4:
   1. Is this a linear equation? Why or why not?
   2. Does this equation represent a relationship?

Solutions

1. A positive slope indicates that the variables increase and decrease together.
2. A line with no slope is a horizontal line, since the only defined variable is the output. No matter what value is given for the explanatory variable, the response is the same.
3. Answers:
   1. The slope, m, is 5.2.
   2. The y-intercept, b, is 7.
   3. Since this line has a positive slope, y increases as x increases.
4. Answers:
   1. No, the explanatory variable is squared, this graph would form a parabola.
   2. Yes! It is just not a linear relationship.

Example 4

Elaina is curious about the relationship between the weight of a dog and the amount of food it eats. Specifically, she wonders if heavier dogs eat more food, or if age and size factor in. She works at the Humane Society, and does some research. After some calculation, she determines that dog weight and food weight exhibit an r-value of 0.73. What can Elaina say about the relationship, based on her research? What percentage of the increases in food intake can she attribute to weight, according to her research?

Solution

The calculated r-value of 0.73 tells us that Elaina’s data demonstrates a moderate to strong correlation between the variables. Since the coefficient of determination tells us the percentage of changes in the output variable that can be attributed to the input variable, we need to calculate r²: r² = (0.73)² = .5329 Approximately 53% of increases in food intake can be attributed to the linear relationship between food intake and the weight of the dog, suggesting that other factors, perhaps age and size, are also involved.

Example 5

Tuscany wonders if barrel racing times are related to the age of the horse. Specifically, she wonders if older horses take longer to complete a barrel racing run. As a member of the Pony Club, she does some research, and determines that horse age to barrel run time exhibits an r-value of 0.52. What can Tuscany say about horse age vs barrel race time, according to her research?

Solution

Tuscany’s research suggests that there is a moderate to weak correlation between horse age and barrel run time. In other words, the research suggests that (0.52)² = .27 = 27% of the differences between barrel run times could be attributable to the linear relationship between barrel run time and the age of the horse.

Example 6

Sayber has collected the following data regarding player score vs age in his favorite online game. He suspects that increased age is not a good indicator of gaming ability. What are the linear correlation coefficient and coefficient of determination values of his data, and how do they support or not support Sayber’s hypothesis?

Table 4
Age	Avg. Player Score
12	5,120
14	6,328
18	7,892
22	7,340
28	6,987
34	7,750
42	5,421

Solution

Let’s use the online calculator at easycalculation.com for this one. I entered the explanatory (Age) and response (Player Score) values into the calculator: The linear correlation coefficient of approximately 0.04 suggests that there is no appreciable linear correlation. The coefficient of determination of 0.0016 suggests that perhaps 0.16% (practically none) of the variability of the player score is dependent on age. Looking at the scores, however, something seems a miss with our findings. The scores suggest that age has no bearing on player score, but look at the graph of the same data: The graph suggests that the youngest and oldest polled players score less than players in late teens to mid-thirties, which seems reasonable. This is an important example of the weakness of using just one indicator of the relationship between two variables. As I noted early in the reading, the r-value is only an indicator of linear correlation, it says nothing at all about other kinds of variable relationships. It is always a good idea to review your data in different ways to evaluate your initial conclusions.

Guided Practice

1. What can you say about the strength of a linear relationship with an r-value of −0.87?
2. What can you say about the level of negative correlation of a relationship if you know the coefficient of determination is 0.82?
3. How much of the variability of y is attributable to x in a relationship with an r-value of 0.76?

Solutions

1. An |r| of > 0.85 indicates a strong linear relationship. The fact that r is negative indicates that as x increases, y decreases.
2. Nothing! The coefficient of determination is r², and therefore always positive. We know that |r|=[latex]\sqrt{.82}\approx{.91}\\[/latex], so this is a strong linear correlation, but we have no idea if it is positive or negative.
3. The coefficient of determination describes the variation in y attributable to x, so we need to find r²: (0.76)2 = .5776. Approximately 57.76% of the change in y-values can be attributed to the change in x.