Activity
For each situation, describe the linear relationship between the variables, based on the correlation coefficient. Make sure to mention whether there is a strong relationship or not as well as whether it is a positive relationship or negative relationship.
Number of steps taken per day and number of kilometers walked per day.
Compare your answer:
Your answer may vary, but here is a sample.
There is a strong, positive linear relationship. An increase in the number of kilometers walked tends to go along with an increase in the number of steps taken per day.
Temperature of a rubber band and distance the rubber band can stretch.
Compare your answer:
Your answer may vary, but here is a sample.
There is a strong, positive linear relationship. An increase in temperature tends to go along with an increase in the distance a rubber band can stretch.
Car weight and distance traveled using a full tank of gas.
Compare your answer:
Your answer may vary, but here is a sample.
There is a strong, negative relationship. An increase in the weight of a car tends to go along with a decrease in the distance it can travel on a full tank of gas.
Average fat intake per citizen of a country and average cancer rate of a country.
Compare your answer:
Your answer may vary, but here is a sample.
There is a moderate positive linear relationship. An increase in the fat intake per citizen of a country tends to go along with an increase in the average cancer rate of a country, but it is not a strong relationship.
Score on science exam and number of words written on the essay question.
Compare your answer:
Your answer may vary, but here is a sample.
There is a weak, positive linear relationship. An increase in the number of words written on the essay question tends to go along with an increase on the score for a science exam, but the relationship is weak.
Average time spent listening to music per day and average time spent watching TV per day.
Compare your answer:
Your answer may vary, but here is a sample.
There is a weak, negative relationship. An increase in the average time spent listening to music per day tends to go along with a decrease in the average time spent watching TV per day, although the relationship is weak.
Are you ready for more?
Extending Your Thinking
A biologist is trying to determine if a group of dolphins is a new species of dolphin or if it is a new group of individuals within the same species of dolphin. The biologist measures the width (in millimeters) of the largest part of the skull, zygomatic width, and the length (in millimeters) of the snout, rostral length, of 10 dolphins from the same group of individuals.
| , Rostral length (mm) | , Zygomatic width (mm) |
|---|---|
|
288 |
147 |
|
247 |
147 |
|
268 |
171 |
|
278 |
177 |
|
258 |
168 |
|
272 |
184 |
|
272 |
161 |
|
258 |
159 |
|
273 |
168 |
|
277 |
166 |
The data appears to be linear and the equation of the line of best fit is and the -value is 0.201.
After checking the data, the biologist realizes that the first zygomatic width listed as 147 mm is an error. It is supposed to be 180 mm. Use technology to find the equation of a line of best fit and the correlation coefficient for the corrected data. What is the equation of the line of best fit and the correlation coefficient?
Compare your answer:
and
Compare the new equation of the line of best fit with the original. What impact did changing one data point have on the slope, -intercept, and correlation coefficient on the line of best fit?
Compare your answer:
Your answer may vary, but here is a sample.
The slope increased and the correlation coefficient got closer to 1. The -intercept also change significantly.
Why do you think that weak positive association became a moderately strong association? Be prepared to show your reasoning.
Compare your answer:
Your answer may vary, but here is a sample.
I think it became moderately strong because the data point that was changed did not follow the roughly linear pattern of the other 9 data points, and when it was changed, it did follow the linear pattern more closely.
For questions 4—6, use technology to change the -value for the first and second entries in the table.
How does changing each point's -value impact the correlation coefficient?
Compare your answer:
Your answer may vary, but here is a sample.
Changing the points to numbers well above or below the existing linear trend in the data makes the correlation coefficient get closer to zero.
Can you change two values to get the correlation coefficient closer to 1? Use data to support your answer.
Compare your answer:
Your answer may vary, but here is a sample.
For example: I managed to get the -value to 0.86 by changing the 2 values that appeared furthest from the line of best fit, and , to values that are very close to the existing line of best fit, and .
By leaving (288,180), can you change a value to get the relationship to change from a positive one to a negative one? Use data to support or refute your answer.
Compare your answer:
Your answer may vary, but here is a sample.
By changing to , I was able to change the relationship to a negative one because the line of best fit had a negative slope, .
Self Check
In the graph below, which best describes the scatter plot of points?
-
, positive correlation
-
, weak, negative correlation
-
, strong, negative correlation
-
, positive, weak, correlation
Additional Resources
Determine the Strength of a Linear Relationship
Finding a Correlation Coefficient
- Calculate the correlation coefficient for cricket-chirp data in the table below.
- Tell if the linear relationship is positive or negative and if it is strong, weak, or moderate.
| Chirps | 44 | 35 | 20.4 | 33 | 31 | 35 | 18.5 | 37 | 26 |
| Temperature | 80.5 | 70.5 | 57 | 66 | 68 | 72 | 52 | 73.5 | 53 |
- Using graphing technology, enter the chirps in the -column of the table and the temperature in the -column of the table. The correlation coefficient is .
- This value is very close to 1, which suggests a strong, positive, linear relationship.
Try it
Try It: Determine the Strength of a Linear Relationship
Calculate the correlation coefficient for data in the table below. Tell if the linear relationship is positive or negative and if it is strong, weak, or moderate.
| 21 | 25 | 30 | 31 | 40 | 50 | |
| 17 | 11 | 2 | -1 | -18 | -40 |
Compare your answer:
, The relationship is a strong and negative linear relationship.