Section 2.6: Two Quantitative Variables: Linear Regression ...

Section 2.6: Two Quantitative Variables: Linear Regression Objectives

1) Least squares regression line ? use calculator to find it 2) When can we use it for predictions? 3) Interpret the slope and intercept 4) Calculate residuals and show them on the scatterplot 5) Outliers (influential observations) 6) Cautions with regression

a. Extrapolation b. Always plot the data (why?) c. Effect of outliers

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Section 2.6: Two Quantitative Variables: Linear Regression

Data 2.10 ? Example 2.35 - page 111 ? from section 2.5 ? "Cricket Chirps and Temperature"

1) Here are the data from page 111

Chirps per

81

97

103

123

150

182

195

minute (x)

Temperature 54.5

59.5

63.5

67.5

72.0

78.5

83.0

F (y)

a) Label axes using the temperature as response variable, Y, and

b)

the chirps per minute as explanatory variable, X. Create a

TO obtain the correlation coefficient we need

scatterplot using the ordered pairs in the order (chirps per to TURN DIAGNOSTICS ON

minute, temperature)

In the TI-84 graphing calculator:

1. Press 2nd 0 (CATALOG)

2. Scroll with the arrow key down until you

see DIAGNOSTICS ON

3. Press ENTER twice

c) Calculate the regression line and the correlation coefficient.

d) Is there a linear association between the variables? Will you classify it as weak, moderate or strong? Explain.

e) Interpret the slope in context.

f) Use the equation for predictions If you listen and hear crickets chirping about 140 times per minute, what is your best guess at the outside temperature? Calculate and label the point on the graph.

g) For a temperature of 75 ?F, what is the expected number of chirps per minute?

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Section 2.6: Two Quantitative Variables: Linear Regression

2) Cricket problem, continued ? Calculating residuals

Chirps per

81

97

103

123

150

182

195

minute

Temperature 54.5

59.5

63.5

67.5

72.0

78.5

83.0

F

Predicted

values (use

line)

Residuals

a) One of the cases in the cricket dataset is 103 chirps per minute and 63.5oF. Use the linear model to predict the temperature for 103 chirps per minute.

b) Find the residual for this case. Show the residual on the graph on the previous page.

c) Do the same for the ordered pair (81, 54.5)

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Section 2.6: Two Quantitative Variables: Linear Regression Below you find the scatter plot for the "Presidential rating" problem from section 2.5

a) Use the data from section 2.5 to find the line of best fit. Do not include data for Obama. b) Draw a rough sketch of the line in the scatterplot c) Calculate the residual for the president that had the largest margin of victory.

a. Draw the residual in the plot d) Predict the approval rating of Obama who had a margin of victory of ______ (look at the notes for 2.5)

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Section 2.6: Two Quantitative Variables: Linear Regression

3) Using Neck Circumference to Predict Body Fat The regression line for predicting body fat percent using neck circumference is

(a) What body fat percent does the line predict for a person with a neck circumference of 35 cm? Of 40 cm?

(b) Interpret the slope of the line in context.

(c) One of the men in the study had a neck circumference of 38.7 cm and a body fat percent of 11.3. Find the residual for this man.

4) An Outlier - Jogging Times - The Table gives the times (in minutes) for five races in which two joggers participated. Jogging times (min)

Jogger A Jogger B

44

48

45

49

43

38

48

40

45

50

(a) Construct a scatterplot of the race times. (b) Use technology to find the correlation coefficient and the regression line.

(c) Should we use the regression line for predictions? Explain.

(d) A sixth race is held on a very windy day, and jogger A takes 50 minutes while jogger B takes a whole hour to complete the race. Recalculate the correlation with this point added.

(e) Compare correlations from parts (b) and (d). Did adding the results from the windy day have an effect on the relationship between the two joggers?

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