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Chapter 3: Q.26 (page 162)

Strong association but no correlation The gas mileage of an automobile first increases and then decreases as the speed increases. Suppose that this relationship is very regular, as shown by the following data on speed (miles per hour) and mileage (miles per gallon). Make a scatterplot of mileage versus speed.

The correlation between speed and mileage isr=0. Explain why the correlation is 0 even though there is a strong relationship between speed and mileage.

Short Answer

Expert verified

The linear correlation coefficient is 0, because there is no LINEAR pattern in the scatterplot and the linear correlation coefficient only measures a linear pattern.

Step by step solution

01

Given Information

The speed and mileage have ar=0correlation.

02

Explanation 

The horizontal axis represents speed, while the vertical axis represents mileage.

03

Conclusion

The correlation is set to 0, indicating that there is no linear correlation.

The speed and mileage have a perfect quadratic relationship.

Because there is no LINEAR pattern in the scatterplot and the linear correlation coefficient only assesses a linear pattern, the linear correlation coefficient is 0.

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Most popular questions from this chapter

The graph at the top right plots the gas mileage (miles per gallon) of various cars from the same model year versus the weight of these cars in thousands of pounds. The points marked with red dots correspond to cars made in Japan. From this plot, we may conclude that

(a) there is a positive association between weight and gas mileage for Japanese cars.

(b) the correlation between weight and gas mileage for all the cars is close to 1.

(c) there is little difference between Japanese cars and cars made in other countries.

(d) Japanese cars tend to be lighter in weight than other cars.

(e) Japanese cars tend to get worse gas mileage than other cars.

Do heavier people burn more energy? Metabolic rate, the rate at which the body consumes energy, is important in studies of weight gain, dieting, and exercise. We have data on the lean body mass and resting metabolic rate for 12women who are subjects in a study of dieting. Lean body mass, given in kilograms, is a person’s weight leaving out all fat. Metabolic rate is measured in calories burned per 24hours. The researchers believe that lean body mass is an important influence on metabolic rate.

(a) Make a scatterplot on your calculator to examine the researchers’ belief.

(b) Describe the direction, form, and strength of the relationship.

Least-squares idea The table below gives a small set of data. Which of the following two lines fits the data better: y^=1−xory^=3−2x? Make a graph of the data and use it to help justify your answer. (Note: Neither of these two lines is the least-squares regression line for these data.)

Data on dating Refer to Exercise 20.

(a) How would r change if all the men were 6 inches shorter than the heights given in the table? Does the correlation tell us if women tend to date men taller than themselves?

(b) If heights were measured in centimeters rather than inches, how would the correlation change? (There are 2.54centimeters in an inch.)

Joan is concerned about the amount of energy she uses to heat her home. The graph below plots the mean number of cubic feet of gas per day that Joan used each month against the average temperature that month (in degrees Fahrenheit) for one heating season.

(a) Does the plot show a positive or negative association between the variables? Why does this make sense?

(b) What is the form of the relationship? Is it very strong? Explain your answers.

(c) Explain what the point at the bottom right of the plot represents.
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