11. Correlation

4. Give examples of two variables that have perfect positive linear correlation and two variables that have perfect negative linear correlation.

Given a scatterplot showing a strong negative linear relationship between two variables, which of the following is most likely the correlation coefficient for the data set?

For which of the following scatterplots is the correlation between $x$ and $y$ closest to $0$?

Which of the following scatter plots suggests a linear relationship between two variables?

[NW] Television Stations and Life Expectancy Based on data obtained from the CIA World Factbook, the linear correlation coefficient between the number of television stations in a country and the life expectancy of residents of the country is 0.599. What does this correlation imply? Do you believe that the more television stations a country has, the longer its population can expect to live? Why or why not? What is a likely lurking variable between number of televisions and life expectancy?

[DATA] Crickets make a chirping noise by sliding their wings rapidly over each other. Perhaps you have noticed that the number of chirps seems to increase with the temperature. The following data list the temperature (in degrees Fahrenheit) and the number of chirps per second for the striped ground cricket.

c. Calculate the linear correlation coefficient between temperature and chirps per second.

In Exercise 23, add data for a child who is 6 years old and has a vocabulary of 900 words. Describe how this affects the correlation coefficient r.

In Exercises 5 and 6, use the scatterplot to find the value of the rank correlation coefficient and the critical values corresponding to a 0.05 significance level used to test the null hypothesis of . Determine whether there is a correlation.

Altitude and Temperature Shown below is a scatterplot of altitudes (thousands of feet) and outside air temperatures (degrees Fahrenheit) recorded by the author during a Delta flight from New Orleans to Atlanta.

Testing for a Linear Correlation

In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of α = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Powerball Jackpots and Tickets Sold Listed below are the same data from Table 10-1 in the Chapter Problem, but an additional pair of values has been added in the last column. Is there sufficient evidence to conclude that there is a linear correlation between lottery jackpot amounts and numbers of tickets sold? Comment on the effect of the added pair of values in the last column. Compare the results to those obtained in Example 4.

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Interpreting r

In Exercises 5–8, use a significance level of α = 0.05 and refer to the accompanying displays.

Bear Weight and Chest Size Fifty-four wild bears were anesthetized, and then their weights and chest sizes were measured and listed in Data Set 18 “Bear Measurements” in Appendix B; results are shown in the accompanying Statdisk display. Is there sufficient evidence to support the claim that there is a linear correlation between the weights of bears and their chest sizes? When measuring an anesthetized bear, is it easier to measure chest size than weight? If so, does it appear that a measured chest size can be used to predict the weight?

Testing for Rank Correlation

In Exercises 7–12, use the rank correlation coefficient to test for a correlation between the two variables. Use a significance level of α = 0.05.

Computers The following table lists quality rankings and costs (dollars) for different brands of laptop computers with 12 in. or 13 in. screens (based on data from Consumer Reports). Lower values of the quality rankings correspond to better computers. Do the more expensive brands appear to have better quality?

The TIMMS Exam The Trends in International Mathematics and Science (TIMMS) is a mathematics and science achievement exam given internationally. On each exam, students are asked to respond to a variety of background questions. For the 41 nations that participated in TIMMS, the correlation between the percentage of items answered in the background questionnaire (used as a proxy for student task persistence) and mean score on the exam was 0.79. Does this suggest there is a linear relation between student task persistence and achievement score? Write a sentence that explains what this result might mean.

Testing for a Linear Correlation

In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of α = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Taxis Using the data from Exercise 15, is there sufficient evidence to support the claim that there is a linear correlation between the distance of the ride and the fare (cost of the ride)?

In Problems 3–6, use the results in the table to (b) determine the linear correlation between the observed values and expected z-scores, (c) determine the critical value in Table VI to assess the normality of the data.

What does it mean when rs is equal to 1? What does it mean when rs is equal to ? What does it mean when rs is equal to 0?