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# Linear Regression and Correlation: Teacher's Guide

Summary: This module is the complementary teacher's guide for the "Linear Regression and Correlation" chapter of the Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Entire courses are given on linear regression and correlation. This chapter serves as an introduction to the topics.

It helps to review the equation of a line. We use aa size 12{a} {} for the yy size 12{y} {}-intercept and bb size 12{b} {} for the slope. The line has the form: y=a+bxy=a+bx size 12{y=a+ ital "bx"} {}

## Example 1

### Problem 1

Have the students plot a line by eye using the following data. The independent variable x x size 12{x} {} represents the size of a color television screen in inches at Anderson's and y y alignl { stack { size 12{y} {} # {} } } {} represents the sales price in dollars.

Table 1
x x size 12{x} {} 9 20 27 31 35 40 60
y y size 12{y} {} 147 197 297 447 1177 2177 2497

Ask them what they got for the slope and for the y-intercept. Make comparisons. This exercise should point out how difficult it is to get an accurate line of best fit and how many lines "seem" to fit the data. (This data is taken from the exercises.)

#### Solution

For the data above, use either a calculator or a computer and calculate the least squares or best fit line. Look at the scatter plot first. Ask the students if their "by eye" line looks like the calculated one. Explain the correlation coefficient and then check if the correlation coefficient is significant by comparing it to the correct entry in 95% CRITICAL VALUES OF THE SAMPLE CORRELATION COEFFICIENT Table at the end of the reading.

If you use the TI-83/84 series, enter the data into two lists first. Then plot the data points on the calculator. First set up the stat plot (2nd STAT PLOT). Then press ZOOM 9 to see the plot. To do the linear regression, go to the LinReg ( a + bx ) ( a + bx ) size 12{ $$a+ ital "bx"$$ } {} function in STAT CALC. Enter the lists. At this time, you could also enter a y-variable after the lists (after you enter the lists, enter a comma and then press VARS Y-VARS Function Y1). Press ENTER to see the linear regression. When you press GRAPH, the line will plot.

Line of best fit: yhat=745.2420+54.7557xyhat=745.2420+54.7557x size 12{ ital "y-hat"= - "745" "." "2420"+"54" "." "7557"x} {}.

Explain "predicting" (or forecasting) and have them predict the sales price of a 45 inch screen color TV. Have them predict the cost for a mini 5 inch color TV. (The answer is negative.) Discuss that the line is only valid from the lowest to the highest x x size 12{x} {} - values.

## Example 2

### Problem 1

Have the students follow the "outlier" example in the text and (just once!) do the calculations for finding an outlier. Have them fill in the table below.

 x x size 12{x} {} y y size 12{y} {} y − yhat y − yhat size 12{y - ital "yhat"} {} ∣ y − yhat ∣ ∣ y − yhat ∣ size 12{ lline y - ital "yhat" rline } {} ( ∣ y − yhat ∣ ) 2 ( ∣ y − yhat ∣ ) 2 size 12{ $$lline y - ital "yhat" rline$$ rSup { size 8{2} } } {}

Find: 7(yyhat)2=SSE7(yyhat)2=SSE size 12{ Sum rSub { size 8{7} } { $$lline y - ital "yhat" rline }$$ rSup { size 8{2} } = ital "SSE"} {}

Find s=SSEn2s=SSEn2 size 12{s= sqrt { { { ital "SSE"} over {n - 2} } } } {}

n=n= size 12{n={}} {}the total number of data values (7 for this problem)

ss size 12{s} {} is the standard deviation of the yyhatyyhat size 12{ lline y - ital "yhat" rline } {} values

Multiply ss size 12{s} {} by 1.9: (1.9)(s)=(1.9)(s)= size 12{ $$1 "." 9$$ $$s$$ ={}} {}_______

Compare each yyhatyyhat size 12{ lline y - ital "yhat" rline } {} to (1.9)(s)(1.9)(s) size 12{ $$1 "." 9$$ $$s$$ } {}.

If any yyhatyyhat size 12{ lline y - ital "yhat" rline } {} is at least (1.9)(s)(1.9)(s) size 12{ $$1 "." 9$$ $$s$$ } {}, then the corresponding point is an outlier. (None of the points is an outlier.)

## Assign Practice

Have the students do the Practice collaboratively in class.

## Assign Homework

Assign Homework. Suggested homework: 1, 3, 5, 9, 13, 15 (a - f only if you use the calculator), 21 - 25.

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