Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the corresponding height (m) of the soccer ball.

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c. What horrible mistake would be easy to make if the analysis is conducted without a scatterplot?

Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the corresponding height (m) of the soccer ball.

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b. Based on the result from part (a), what do you conclude about a linear correlation between time and height?

Least-Squares Property According to the least-squares property, the regression line minimizes the sum of the squares of the residuals. Refer to the jackpot/tickets data in Table 10-1 and use the regression equation y^ = -10.9 + 0.174x that was found in Examples 1 and 2 of this section.

a. Identify the nine residuals.

Notation The author conducted an experiment in which the height of each student was measured in centimeters and those heights were matched with the same students’ scores on the first statistics test.

a. For this sample of paired data, what does r represent, and what does represent?

Effects of an Outlier Refer to the Minitab-generated scatterplot given in Exercise 9 of Section 10-1

a. Using the pairs of values for all 10 points, find the equation of the regression line.

Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:

y^ = 58.9 - 0.00749x, where x represents weight.

a. What does the symbol y^ represent?

Explore!

Exercises 11 and 12 provide two data sets from “Graphs in Statistical Analysis,” by F. J. Anscombe, the American Statistician, Vol. 27. For each exercise,

a. Construct a scatterplot.

Effects of Clusters Refer to the Minitab-generated scatterplot given in Exercise 10 of Section 10-1.

a. Using the pairs of values for all 8 points, find the equation of the regression line.

Clusters Refer to the Minitab-generated scatterplot. The four points in the lower left corner are measurements from women, and the four points in the upper right corner are from men.

a. Examine the pattern of the four points in the lower left corner (from women) only, and subjectively determine whether there appears to be a correlation between x and y for women.

Variation and Prediction Intervals

In Exercises 17–20, find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. In each case, there is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions.

Altitude and Temperature Listed below are altitudes (thousands of feet) and outside air temperatures (°F) recorded by the author during Delta Flight 1053 from New Orleans to Atlanta. For the prediction interval, use a 95% confidence level with the altitude of 6327 ft (or 6.327 thousand feet).

Notation The author conducted an experiment in which the height of each student was measured in centimeters and those heights were matched with the same students’ scores on the first statistics test.

b. Without doing any research or calculations, estimate the value of r.

Outlier Refer to the accompanying Minitab-generated scatterplot.

b. After identifying the 10 pairs of coordinates corresponding to the 10 points, find the value of the correlation coefficient r and determine whether there is a linear correlation.

Least-Squares Property According to the least-squares property, the regression line minimizes the sum of the squares of the residuals. Refer to the jackpot/tickets data in Table 10-1 and use the regression equation y^ = -10.9 + 0.174x that was found in Examples 1 and 2 of this section.

b. Find the sum of the squares of the residuals.

Notation The author conducted an experiment in which the height of each student was measured in centimeters and those heights were matched with the same students’ scores on the first statistics test.

c. Does r change if the heights are converted from centimeters to inches?

Sum of Squares Criterion In addition to the value of another measurement used to assess the quality of a model is the sum of squares of the residuals. Recall from Section 10-2 that a residual is (the difference between an observed y value and the value predicted from the model). Better models have smaller sums of squares. Refer to the U.S. population data in Table 10-7.

c. Verify that according to the sum of squares criterion, the quadratic model is better than the linear model.

Notation Using the weights (lb) and highway fuel consumption amounts (mi/gal) of the 48 cars listed in Data Set 35 “Car Data” of Appendix B, we get this regression equation:

y^ = 58.9 - 0.00749x, where x represents weight.

c. What is the predictor variable?

Clusters Refer to the Minitab-generated scatterplot. The four points in the lower left corner are measurements from women, and the four points in the upper right corner are from men.

Find the value of the linear correlation coefficient using all eight points. What does that value suggest about the relationship between x and y?

Exercises 1–10 are based on the following sample data consisting of costs of dinner (dollars) and the amounts of tips (dollars) left by diners. The data were collected by students of the author.

Predictions Repeat the preceding exercise assuming that the linear correlation coefficient is r = 0.132.

Comparing Two Means Treating the data as samples from larger populations, test the claim that there is a significant difference between the mean of presidents and the mean of popes.

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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