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Algebra 1

3.5.3 Using Correlation Coefficient to Describe Relationships between Two Variables

Algebra 13.5.3 Using Correlation Coefficient to Describe Relationships between Two Variables

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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.

1.

Number of steps taken per day and number of kilometers walked per day. r=0.92r=0.92

2.

Temperature of a rubber band and distance the rubber band can stretch. r=0.84r=0.84

3.

Car weight and distance traveled using a full tank of gas. r=0.86r=0.86

4.

Average fat intake per citizen of a country and average cancer rate of a country. r=0.73r=0.73

5.

Score on science exam and number of words written on the essay question. r=0.28r=0.28

6.

Average time spent listening to music per day and average time spent watching TV per day. r=0.17r=0.17

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.

xx, Rostral length (mm) yy, 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  𝑦=0.201𝑥+110.806𝑦=0.201𝑥+110.806   and the  rr -value is 0.201.

1.

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?

2.

Compare the new equation of the line of best fit with the original. What impact did changing one data point have on the slope,   yy -intercept, and correlation coefficient on the line of best fit?

3.

Why do you think that weak positive association became a moderately strong association? Be prepared to show your reasoning.

For questions 4—6, use technology to change the yy-value for the first and second entries in the table.

4.

How does changing each point's   yy-value impact the correlation coefficient?

5.

Can you change two values to get the correlation coefficient closer to 1? Use data to support your answer.

6.

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.

Self Check

In the graph below, which best describes the scatter plot of points?

SCATTER PLOT THAT CAN BE APPROXIMATED BY A LINE, WHERE THE \(y\)-values DECREASE AS THE \(x\)-values INCREASE

  1. r = 0.24 , positive correlation
  2. r = 0.24 , weak, negative correlation
  3. r = 0.90 , strong, negative correlation
  4. r = 0.90 , positive, weak, correlation

Additional Resources

Determine the Strength of a Linear Relationship

Finding a Correlation Coefficient

  1. Calculate the correlation coefficient for cricket-chirp data in the table below.
  2. 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
  1. Using graphing technology, enter the chirps in the xx-column of the table and the temperature in the yy-column of the table. The correlation coefficient is r=0.9509r=0.9509.
  2. 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.

xx 21 25 30 31 40 50
yy 17 11 2 -1 -18 -40

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