How to Get a Function from Data
The procedure described here is for use with TI-83/84
calculators. [If you are using another calculator, read your
instructions on how to perform "regression".]
This data represents accumulated profit (y in $100) per hour (x) at a concession stand.
Here's how the calculator can find the slope and the
y -intercept of a line to predict y
from x.
First input the data into your TI-83/4. To do this, press
STAT ENTER and clear out
columns L1 and L2. Put your x- values in L1 and your y-values in L2. To plot the
three
pairs of points, press 2ND STATPLOT ENTER and turn on Plot 1. Choose the scatter
plot
- the first graph type. Make sure xLIST is L1 and yLIST is L2. Next press WINDOW
and
change Xmin to 0, Xmax to 10, Ymin to 0 and Ymax to 5. Press graph and your
graph
should appear.
Now to get the linear regression equation expressing y in
terms of x, press STAT, choose
the CALC menu and select 4: LinReg(ax+b). Here, a represents the slope of the
line and
b represents the intercept. (Menu number 8 is also a linear model , but it calls
a the
intercept and b the slope .)
Once the command is on the home screen, press 2ND 1 (L1) [
, ] 2ND 2 (L2) to ensure that
your calculator is looking at the correct lists. Your screen should look like
this :
LinReg (ax+b) L1,L2
Then press ENTER. Record the slope, intercept, line equation and value of r here:
a =_______ b =_______ ; y = ___________ x + __________; r =___________
(If you do not see the correlation coefficient ( r ) ,
press 2ND 0 (CATALOG), scroll down and select
DiagnosticOn. Repeat the calculation.)
Now suppose you want to predict the profit after 4.5
hours. Substitute 4.5 into the
equation for x and compute the corresponding value for y. Calculating by hand is
easy,
but your calculator can also be used. One way is as follows:
Press Y= and clear any functions you may find in the list. Make sure your cursor
is on
Y1. Press VARS 5 to get to the EQuation menu.
Select ReqEQ. The most recently
computed regression equation (which is your line equation) should appear in the
function
list as Y1. It can now be graphed, a table of values can be computed, etc. Our
purpose,
though, is to evaluate the function at a specific value of x using the Y-VARS
feature.
Starting from the home screen, select VARS then Y-VARS and
press ENTER twice to
copy Y1 to the screen. Press ( 4.5 ) so that the screen contains Y1(4.5).
Then press ENTER. Write your answer here:
When x = 4.5 hours, y = _____ hundred dollars
To graph the linear regression function you stored in the
function list, simply press
GRAPH. The line should graph through the points graphed earlier.
The points in this example fit exactly on the line. So if
you put, say, 3 in for x, you'll get
exactly 0.5. If the points don't fit a line exactly, the same calculator entry
procedure is
used, but the line you get cannot fit all the points exactly. It will, however,
be the "best"
fit in the " Least Squares sense". (If you care to know what that means, ask. It
takes a few
minutes of explanation; it is not necessary for our purpose today, but is
interesting.)
Now let's apply our knowledge and extend it a bit.
Is the Globe Warming?
The Goddard Institute for Space Studies.
has recorded mean temperatures for the decades 1880 - the present. Here's their
data,
coded so that 1800 is "year 0". That means "8" is the 1880s, "12" is the 1920s,
etc.
Code for Decade | Degrees Celsius | |
Decade | X | Y |
1880-1889 | 8 | 13.82 (This is 56.876° F.) |
1890-1899 | 9 | 13.69 (°F = 1.8 °C + 32) |
1900-1909 | 10 | 13.74 |
1910-1919 | 11 | 13.79 |
1920-1929 | 12 | 13.91 |
1930-1939 | 13 | 14.02 |
1940-1949 | 14 | 14.05 |
1950-1959 | 15 | 13.98 |
1960-1969 | 16 | 13.94 |
1970-1979 | 17 | 14.02 |
1980-1989 | 18 | 14.26 |
1990-1999 | 19 | 14.40 |
2000-2005 | 20 | 14.62 (This is 58.316° F.) |
Using your calculator, (a) find the Least Squares Line
that predicts the average
temperature (Y) when given the coded value for a decade (X). Put your answer
here:
Y = _____________ X + ______________. Also record r^2: ________
(b) Use your function to approximate the average
temperature in the decade of the
2060s. (If you are a freshman, this will probably be the first decade you are
retired.)
Y(_____________) = ______________.
Now graph your points and your function. Note that the
points "bend" but the line, of
course, is straight. Recall from algebra that the graph of a quadratic function
can bend.
Your calculator can easily produce a quadratic function (Y = aX^2 + bX + c). All
you
have to do is choose 5:QuadReg from the STAT|CALC menu (instead of
4:LinReg(ax+b)).
(c) Re-run the regression choosing the quadratic model.
Record your results here:
Y = _______ X^2 +________ X + ________. Again record r^2: ________
Graph your new model. Note the fit. Is it better than the line? ______
(d) And once again, predict the mean temperature in the
decade of the 2060s:
Y(_____________) = ______________.
(e) What major world event occurred in the late '30s and
early '40s?
_____________________________________
Additional information : The value r^2 (which is just the
square of the correlation
coefficient "r", and is called the "coefficient of determination") tells us, in
layman's
terms , the proportion of variation in Y that is described by the variation in X.
More
simply, if r^2 is closer to 1, the curve "fits" the points better. We'll discuss
this more in
class. For now, note the two values of r^2.
Very important assignment:
In a paragraph or two, tell which model above (the
quadratic model or the linear model)
you believe to be the more credible. Explain your choice. If you don't believe
either tells
the story of the globe's warming, explain your belief. In either case, base your
explanation on DATA, not just your gut feelings. An extra credit homework grade
will
be given if the writing assignment is word-processed.
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