Download Intro to Regressions

Intro to Regressions – Statistics
Name ____________________________ 12-1
Regressions are frequently used to understand a relationship between two variables. We can then use this
mathematical model (equation) to make predictions (both interpolations and extrapolations). We’ll work
with linear, exponential, power, logarithmic, and sinusoidal regressions this year.
Linear Regressions
1) A pediatrician would like to determine the relationship between infant female weights versus age. The
pediatrician studies 100 newborn girls and finds their average weight at the end of 3 month intervals. The
data is shown below.
Age
(months)
Average Weight
(pounds)
0
3
6
9
12
15
7.2
11.8
15.5
18.2
21.5
24.8
a) Find the equation of the linear regression of this data,
rounding all coefficients to the nearest tenth.
b) Using the equation found in part a, predict the weight of a
baby girl after 10 months. Round your answer to the
nearest tenth of a pound.
Plot points and sketch equation.
The use of a model to predict outputs when the input is within the range of the known data is
called interpolation. Interpolation tends to be fairly accurate.
c) Using the equation found in part a, predict the weight of a baby girl after 2 years. Round your answer
to the nearest tenth of a pound.
The use of a model to predict outputs when the input is outside the range of the known data is
called extrapolation. Models are most helpful when they can be used to extrapolate, but tend to
be less accurate.
Exponential Regressions
2) If a cup of coffee is left on a countertop, it will cool off slowly. The following table shows the
temperature of a cup of coffee sitting for 50 minutes.
Time
Temp
˚F
0
5
10
15
20
176.81 162.91 146.34 135.18 126.28
25
118.8
30
35
40
112.53 107.11 102.72
45
50
100.1
96.44
a) Find the equation of the exponential regression for this data,
rounding all values to the nearest thousandth.
b) Use this equation to predict the temperature of the coffee, to
the nearest hundredth, after 32 minutes. Is this an example of
interpolation or extrapolation? Explain.
Plot points and sketch equation.
c) Use this equation to predict the temperature of the coffee after 1 hour. Is this an example of
interpolation or extrapolation? Explain.
d) Will this equation perfectly model this data? Explain.
Power Regressions
3) The accompanying table shows the number of new cases reported by the Nassau and Suffolk County
Police Crime Stoppers program for the years 2000 through 2002.
Year (x)
2000
2001
2002
New Cases (y)
457
369
353
a) If x = 1 represents the year 2000, and y represents the
number of new cases, find the equation of the best fit using a
power regression, rounding all values to the nearest
thousandth.
b) Using this equation, find the estimated number of new cases, to the nearest whole number, for the
year 2007.
Logarithmic Regressions
4) The data below show the average growth rates of 12 Weeping Higan cherry trees planted in Washington,
D.C. At the time of planting, the trees were one year old and were all 6 feet in height.
Age of Tree
(in years)
1
2
3
4
5
6
7
8
9
10
11
Height
(in feet)
6
9.5
13
15
16.5
17.5
18.5
19
19.5
19.7
19.8
a) Determine a logarithmic regression model
equation to represent this data, rounding
coefficients to the nearest thousandth..
Plot points and sketch equation.
b) Interpolate: Using the equation, what was the average height of the trees at one and one-half years of
age? (to the nearest tenth of a foot)
c) Extrapolate: Using the equation, what is the predicted average height of the trees at 20 years of age?
Is this prediction realistic? (answer to the nearest tenth of a foot)
d) Based upon your observations of this data, what would you predict to be the average height of a
mature Higan cherry tree, to the nearest foot?
e) If the average height of the trees is 10 feet, what is the age of the trees to the nearest tenth of a year?
Sinusoidal Regressions
5) When two species interact in a predator /prey relationship, the population of both species tend to vary in
a sinusoidal fashion. In a certain Midwestern county, the main food source for barn owls consists of field
mice and other small mammals. The table below gives the population of barn owls in this county every
July 1 over a 12-year period.
a) Find a sine curve that models the data, rounding all numbers to the nearest ten thousandths.
b) Predict the owl population to the nearest whole number after 20 years.
Plot points and sketch equation.
Calculator Help for Regressions
To enter data:
To find regression equation:
(EDIT)
(CALC)
(LinReg)
or whichever regression you choose
(L1)
(L2)
<Enter data in L1 and L2>
To plot points on scatter plot:
(STAT PLOT)
(PLOT ON)
Write down equation following rounding
directions.
Put equation into Y= to see “best fit curve” on
plotted points.
To find r (correlation coefficient):
(ZoomStat)
(window may need adjusting)
Turn Diagnostics on before finding the linear
equation.
(Catalog)
(D)
for diagnostics
(scroll down til
DiagnosticsON)