Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Priya takes note of the distance the car drives and the time it takes to get to the destination for many trips.

Distance (mi) ( $x$ )	Travel time (min) ( $y$ )
2	4
5	7
10	11
10	15
12	16
15	22
20	23
25	25
26	28
30	36
32	35
40	37
50	51
65	70
78	72

1.

Distance is one factor that influences the travel time of Priya’s car trips. What are some other factors?

Your answers may vary, but here are some examples:

Traffic, stop signs, stop lights, refueling, whether the driver is in a hurry, driving on surface streets or interstate.

2.

Which of these factors (including distance) most likely has the most consistent influence for all the car trips? Be prepared to show your reasoning.

Your answers may vary, but here are some examples:

Distance is the most consistent influence since it is always part of the reason a trip will take the time it does. The other factors, like stop lights, are only a factor sometimes since you may hit all green lights one day and all red lights the next day.

3.

Use the graphing tool or technology outside the course. Create the scatter plot and calculate the best fit line using the Desmos tool below.

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Compare your answer:

Your answer may vary, but here is a s sample. $y = 0.93 x + 4.15$

A scatter plot with black dots showing a positive linear relationship between x and y, and a blue line representing the line of best fit passing through the points. Both axes range from 0 to 80.

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4.

What do the slope and $y$ -intercept for the line of best fit mean in this situation?

Your answers may vary, but here are some examples:

The slope represents the number of minutes it takes to drive one additional mile using the linear model. Assuming the linear relationship continues, the $y$ -intercept represents the number of minutes it takes to drive zero miles; this may account for things like time getting ready or warming up the car before driving.

5.

Use technology to find the correlation coefficient for this data. Based on the value, how would you describe the strength of the linear relationship?

Your answers may vary, but here are some examples:

$𝑟 = 0.99$ . Since $r$ is near 1, there is strong positive linear relationship between distance $(x)$ and travel time $(y)$ .

6.

How long do you think it would take Priya to make a trip of 90 miles if the linear relationship continues?

Your answer may vary, but here are some examples:

It would take Priya about 88 minutes. I used the equation $y = 0.93 x + 4.15$ and figured out that $y$ was 87.85 minutes by substituting 90 miles for $x$ .

Self Check

Determine the correlation coefficient of the data in the table below to 3 decimal places and interpret the value.

$x$	8	15	26	31	56
$y$	23	41	53	72	103

$r=-0.987$ ; there is
$r=0.987$ ; there is a strong positive relationship.
$r=0.987$ ; there is a weak positive relationship.
$r = -0.987$ ; there is a weak negative relationship.

Additional Resources

Calculating and Interpreting Correlation Coefficients

Properties of the correlation coefficient:

Property 1: The sign of $r$ (positive or negative) corresponds to the direction of the linear relationship.
Property 2: A value of $r = + 1$ indicates a perfect positive linear relationship, with all points in the scatter plot falling exactly on a straight line.
Property 3: A value of $r = - 1$ indicates a perfect negative linear relationship, with all points in the scatter plot falling exactly on a straight line.
Property 4: The closer the value of $r$ is to $+ 1$ or $- 1$ , the stronger the linear relationship.

Using the data of shoe length in inches and height in inches for 10 men, find the correlation coefficient of the data with graphing technology.

$x$ (Shoe length) inches	$y$ (Height) inches
12.6	74
11.8	65
12.2	71
11.6	67
12.2	69
11.4	68
12.8	70
12.2	69
12.6	72
11.8	71

The correlation coefficient is $r \approx 0.65$ .

Based on the table, there is a moderate positive linear relationship between shoe length and height.

The table below shows how you can informally interpret the value of a correlation coefficient

If the value of the correlation coefficient is . . .	You can say that . . .
$r = 1.0$	There is a perfect positive linear relationship.
$0.7 \leq r < 1.0$	There is a strong positive linear relationship.
$0.3 \leq r < 0.7$	There is a moderate positive linear relationship.
$0 < r < 0.3$	There is a weak positive linear relationship.
$r = 0$	There is no linear relationship.
$- 0.3 < r < 0$	There is a weak negative linear relationship.
$- 0.7 < r \leq - 0.3$	There is a moderate negative linear relationship.
$- 1.0 < r \leq - 0.7$	There is a strong negative linear relationship.
$r = - 1.0$	There is a perfect negative linear relationship.

Try it

Try It: Calculating and Interpreting Correlation Coefficients

Consumer Reports published a study of fast-food items. The table and scatter plot below display the fat content (in grams) and number of calories per serving for 16 fast-food items.

Scatter plot that shows the relationship between the fat in grams graphed on the x-axis and the calories represented on the y-axis.

Fat(g)	Calories(kcal)
2	268
5	3,003
3	260
3.5	300
1	315
2	160
3	200
6	320
3	420
5	290
3.5	285
2.5	390
0	140
2.5	330
1	120
3	180

What is the correlation coefficient for this data and what does it tell us about the strength and type of relationship represented?

Compare your answer: Your answer many vary, but here is a sample.

The correlation coefficient is $r \approx 0.44$ . By the table, this indicates a moderate, positive linear relationship between fat content and calories.

3.5.2 Finding and Using Correlation Coefficient to Interpret the Strength of Linear Relationships

Activity

Additional Resources

Calculating and Interpreting Correlation Coefficients

Try It: Calculating and Interpreting Correlation Coefficients