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Practice: Linear Regression

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module provides a practice of Linear Regression and Correlation as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Student Learning Outcomes

  • The student will evaluate bivariate data and determine if a line is an appropriate fit to the data.

Given

Below are real data for the first two decades of AIDS reporting. (Source: Centers for Disease Control and Prevention, National Center for HIV, STD, and TB Prevention)

Table 1: Adults and Adolescents only, United States
Year # AIDS cases diagnosed # AIDS deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093

Note:

We will use the columns “year” and “# AIDS cases diagnosed” for all questions unless otherwise stated.

Graphing

Graph “year” vs. “# AIDS cases diagnosed.” Plot the points on the graph located below in the section titled "Plot" . Do not include pre-1981. Label both axes with words. Scale both axes.

Data

Exercise 1

Enter your data into your calculator or computer. The pre-1981 data should not be included. Why is that so?

Linear Equation

Write the linear equation below, rounding to 4 decimal places:

Note:

For any prediction questions, the answers are calculated using the least squares (best fit) line equation cited in the solution.

Exercise 2

Calculate the following:

  • a. a=a= size 12{a={}} {}
  • b. b=b= size 12{b={}} {}
  • c. corr.=corr.= size 12{ ital "corr" "." ={}} {}
  • d. n=n= size 12{n={}} {}(# of pairs)

Solution

  • a. a = -3,448,225 a = -3,448,225 size 12{a="-3,448,225"} {}
  • b. b = 1750 b = 1750 size 12{b="1750"} {}
  • c. corr . = 0 . 4526 corr . = 0 . 4526 size 12{ ital "corr" "." =0 "." "4526"} {}
  • d. n = 22 n = 22 size 12{n="22"} {}

Exercise 3

equation: y^=y^= size 12{ { hat {y}}={}} {}

Solution

y ^ = -3,448,225 y ^ = -3,448,225 size 12{ { hat {y}}="-3,448,225"} {} + 1750 x + 1750 x size 12{+"1750"x} {}

Solve

Exercise 4

Solve.

  • a. When x=1985x=1985 size 12{x="1985"} {}, y^=y^= size 12{ { hat {y}}={}} {}
  • b. When x=1990x=1990 size 12{x="1990"} {}, y^=y^= size 12{ { hat {y}}={}} {}

Solution

  • a. 25,525
  • b. 34,275

Plot

Plot the 2 above points on the graph below. Then, connect the 2 points to form the regression line.

Blank graph with horizontal and vertical axes.

Obtain the graph on your calculator or computer.

Discussion Questions

Look at the graph above.

Exercise 5

Does the line seem to fit the data? Why or why not?

Exercise 6

Do you think a linear fit is best? Why or why not?

Exercise 7

Hand draw a smooth curve on the graph above that shows the flow of the data.

Exercise 8

What does the correlation imply about the relationship between time (years) and the number of diagnosed AIDS cases reported in the U.S.?

Exercise 9

Why is “year” the independent variable and “# AIDS cases diagnosed.” the dependent variable (instead of the reverse)?

Exercise 10

Solve.

  • a. When x=1970x=1970 size 12{x="1970"} {}, y^=y^= size 12{ { hat {y}}={}} {}:
  • b. Why doesn’t this answer make sense?

Solution

  • a. -725

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