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3.8 Practice: linear regression

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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 )

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:

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

Calculate the following:

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

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

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

  • 25,525
  • 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.

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.

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

  • -725

Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
Widad Reply
can you send the book attached ?
Ariel
?
Ariel
What is economics
Widad Reply
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn Reply
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn
what is ecnomics
Jan Reply
this is the study of how the society manages it's scarce resources
Belonwu
what is macroeconomic
John Reply
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
etc
husaini
difference between firm and industry
husaini Reply
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?
Toofiq Reply
explain standard reason why economic is a science
innocent Reply
factors influencing supply
Petrus Reply
what is economic.
Milan Reply
scares means__________________ends resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Zarshad Reply
Why qualify 28 supplies
Milan
what are explicit costs
Nomsa Reply
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
David Reply
economic overview notes
Amahle Reply
identify a demand and a supply curve
Salome Reply
i don't know
Parul
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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Source:  OpenStax, Collaborative statistics (custom online version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11476/1.5
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