AP StatisticsAP StatisticsChapter 3Chapter 3

Practice ProblemsPractice Problems

Linear Regression

A diver is investigating a wreck under the water and has to come up to the surface slowly. The following is a chart detailing his depth from the time he starts ascending.Time (sec)Time (sec) Depth (ft)Depth (ft) Time (sec)Time (sec) Depth (ft)Depth (ft)

00 240240 210210 155155

3030 225225 280280 185185

6060 203203 330330 130130

100100 189189 360360 125125

140140 180180 390390 120120

180180 164164

Linear Regression

1. What is the explanatory variable? What is the response?

2. We wish to perform regression on the data. What are the three conditions that we must check before we attempt to do regression?a)a) The data is quantitativeThe data is quantitative

b)b) The data is linearThe data is linear

c)c) There are no outliersThere are no outliers

Linear Regression

1. Graph the scatterplot and determine if linear regression seems appropriate.a)a) No, regression doesn’t seem appropriate No, regression doesn’t seem appropriate

because there appears to be an outlierbecause there appears to be an outlier

2. Which point is the outlier?a)a) At time 280 seconds, he was at depth of At time 280 seconds, he was at depth of

185 feet.185 feet.

Linear Regression

Determine the equation for the Least Squares Regression Line (LSRL).

Describe the association. There appears to be a strong, negative, There appears to be a strong, negative,

linear association between time and linear association between time and depth, but there appears to be an outlierdepth, but there appears to be an outlier

Determine the correlation. r = -0.932r = -0.932

Eliminate the outlier. What is the new correlation? Why does it change? r = -0.980; it’s stronger because the residuals r = -0.980; it’s stronger because the residuals

are smallerare smaller

)(272.0642.225 timeDepth

Linear Regression

8. Determine the equation for the Least Squares Regression Line (LSRL) without the outlier.

a)

9. Explain the meaning of the slope of the line.a) For every 1 second1 second increase in timetime, our model

predicts an average decreasedecrease of 0.292 feet0.292 feet in depthdepth.

10. Explain the meaning of the b0 in context of this problema) At a time of 0 secondsseconds, our model predicts a

depthdepth of 225.698 feet 225.698 feet.b) Although this makes sense, we know that the diver

began to ascend at a depth of 240 ft.

)(292.0698.225 timeDepth

Linear Regression

11. Describe the relationship between time and depth using r2 to make your description more precise.a) Since r = -0.980, r2 = 0.960

b)b) 96%96% of the variation in the depthdepth can be explained by the approximate linear relationship with the timetime.

Linear Regression

12. Using the modified model, predict the depth of the diver at each of the following times and comment on the confidence of your prediction:a) 2 min. 50 sec.

a)a) ≈ ≈ 176 feet176 feetb) 5 min.

a)a) ≈ ≈ 138 feet138 feetc) 6 min. 30 sec. What is the residual at this time?

a)a) ≈ ≈ 112 feet. The residual is 120 – 112 = 8 112 feet. The residual is 120 – 112 = 8 (or if we use the calculator ≈ 8.27)(or if we use the calculator ≈ 8.27)

d) 10 min.

a)a) ≈ ≈ 50.4 feet50.4 feet

Linear Regression

Using the following summary statistics of a statistics class, determine the LSRL (assume that IQ is the explanatory variable):

Linear Regression

112IQ

107SATS1821SAT

10IQS 893.0r

IQ

SAT

s

srb 1slope

10

107893.0 556.9

xbyb 10

)112(556.918210 b

)(10 IQbSATb 728.750

Using the following summary statistics of a statistics class, determine the LSRL (assume that IQ is the explanatory variable):

Linear Regression

112IQ

107SATS1821SAT

10IQS 893.0r

xbby 10ˆLSRL

556.9 and 728.750 Since 10 bb

IQSAT 556.9728.750

With an LSRL of: Interpret b0

With an IQ of 0, our model predicts an SAT SAT scorescore of 750.728 750.728. This make absolutely no sense. You can’t

have an IQ of 0!

Interpret b1

For every increase of 1 point1 point in IQIQ, our model predicts an average increaseincrease of 9.556 9.556 pointpoint on the SATSAT.

Linear RegressionIQSAT 556.9728.750

With an LSRL of: Interpret rr22

Since r = 0.893, r2 = 0.797 Approximately 80%80% of the variation in SAT SAT

scorescore can be explained by the approximate linear relationship with the IQIQ.

Linear RegressionIQSAT 152.0976.1803

Review Question A researcher uses a regression equation to

predict home heating bills (dollar cost), based on home size (square feet). The correlation between predicted bills and home size is 0.70. What is the correct interpretation of this finding?

a) 70% of the variability in home heating bills can be explained by home size.

b) 49% of the variability in home heating bills can be explained by home size.

c) For each added square foot of home size, heating bills increased by 70 cents.

d) For each added square foot of home size, heating bills increased by 49 cents.

e) None of the above.

The answer is b) since the coefficient of The answer is b) since the coefficient of determination measures the proportion of determination measures the proportion of variation in the dependent variable that is variation in the dependent variable that is predictable from the independent variable. predictable from the independent variable.

Review Question A national consumer magazine reported the A national consumer magazine reported the

following correlations:following correlations:The correlation between car weight and car reliability is -0.30. The correlation between car weight and The correlation between car weight and annual maintenance cost is 0.20.annual maintenance cost is 0.20.

Which of the following statements are true?I. Heavier cars tend to be less reliable.II. Heavier cars tend to cost more to maintain.III. Car weight is related more strongly to reliability than to

maintenance cost. a) I onlyb) II onlyc) III onlyd) I and IIe) I, II, and III

The answer is e) since reliability tends to decrease The answer is e) since reliability tends to decrease as car weight increases, costs tend to increase as as car weight increases, costs tend to increase as car weight increases, and strength increases as car weight increases, and strength increases as correlation gets closer to ±1correlation gets closer to ±1

Review Question In the context of regression analysis, In the context of regression analysis,

which of the following statements are which of the following statements are true?true?

I. When the sum of the residuals is greater than zero, the data set is nonlinear

II. A random pattern of residuals supports a linear model.

III. A random pattern of residuals supports a non-linear model.

a) I onlyb) II onlyc) III onlyd) I and IIe) I and III

The answer is b) since a random pattern of residuals The answer is b) since a random pattern of residuals supports a linear model; a non-random pattern supports a linear model; a non-random pattern supports a non-linear model. The sum of the residuals supports a non-linear model. The sum of the residuals is always zero, whether the data set is linear or is always zero, whether the data set is linear or nonlinear nonlinear