Linear Regression and Correlation: Practice
1.7
2008/06/22 21:22:13 GMT-5
2008/10/27 17:54:41.430 GMT-5
Susan
Dean
deansusan@deanza.edu
Barbara
Illowsky
illowskybarbara@deanza.edu
Susan
Dean
deansusan@deanza.edu
Barbara
Illowsky
illowskybarbara@deanza.edu
Connexions
cnx@cnx.org
elementary
statistics
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 explore the properties of linear regression.
Given
The data below are real. Keep in mind that these are only reported figures. (Source: Centers for Disease Control and Prevention, National Center for HIV, STD, and TB Prevention, October 24, 2003)
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
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
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:
Calculate the following:
- a
a= size 12{a={}} {}
- b
b= size 12{b={}} {}
- c
corr.= size 12{ ital "corr" "." ={}} {}
- d
n= size 12{n={}} {}(# of pairs)
- a
a
=
-3,448,225
size 12{a="-3,448,225"} {}
- b
b
=
1750
size 12{b="1750"} {}
- c
corr
.
=
0
.
4526
size 12{ ital "corr" "." =0 "." "4526"} {}
- d
n
=
22
size 12{n="22"} {}
equation:
y^= size 12{ { hat {y}}={}} {}
y
^
=
-3,448,225
size 12{ { hat {y}}="-3,448,225"} {}
+
1750
x
size 12{+"1750"x} {}
Solve
Solve.
- a
When
x=1985 size 12{x="1985"} {},
y^= size 12{ { hat {y}}={}} {}
- b
When
x=1990 size 12{x="1990"} {},
y^= size 12{ { hat {y}}={}} {}
-
a25082
-
b33,831
Plot
Plot the 2 above points on the graph below. Then, connect the 2 points to form the regression line.
Obtain the graph on your calculator or computer.
Discussion Questions
Look at the graph above.
Does the line seem to fit the data? Why or why not?
Do you think a linear fit is best? Why or why not?
Hand draw a smooth curve on the graph above that shows the flow of the data.
What does the correlation imply about the relationship between time (years) and the number of diagnosed AIDS cases reported in the U.S.?
Why is “year” the independent variable and “# AIDS cases diagnosed.” the dependent variable (instead of the reverse)?
Solve.
- aWhen
x=1970 size 12{x="1970"} {},
y^= size 12{ { hat {y}}={}} {}:
- bWhy doesn’t this answer make sense?
- a
-1164