Regression Analysis
The process of regression analysis allows you to take a series of points and fit an equation to them.
If the scatter plot of the points seems to form a line, you should find a linear regression equation.
If the scatter plot of the points seems to form an exponential curve, you should find an exponential regression
equation.
In other words, the equation you chose to fit the points to should be the appropriate type of equation to match
the behavior of the data.
Looking at a scatter plot of a linear regression and analyzing the correlation coefficient
The correlation coefficient, r, of a linear regression describes the strength of the linear relationship between the
two variables.
The value of r must always line between values of -1 and 1, inclusive.
The closer the r-value is to the value of -1 or 1, the stronger the linear relationship.
You cannot have an r-value greater than 1.
If r = 1, you have a perfect positive correlation - the points lie exactly on a line that has a positive slope.
If r = -1, you have a perfect negative correlation, the points lie exactly on a line that has a negative slope.
If r = 0, there is no reliable correlation between the two variables.
If the line has a negative slope, the r-value must be negative.
If the line has a positive slope, the r-value must be positive.
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To calculate a regression equation
Use your calculator to get the appropriate equation. Be sure to write Y = followed by the General form of the
equation the calculator gives you. Once that general equation is written, then plug in the appropriate values. Be
sure to round, ONLY if the question asks you to, and be sure to round to the appropriate place values. Remember,
tenth is one place after the decimal, then hundredth, thousandth, ten-thousandth, and hundredth-thousandth, etc.
They may also ask you to round to the nearest HUNDRED or THOUSAND or MILLION, etc, be careful not to
mistake those for hundredth, or thousandth, etc.
If you are asked to make a prediction based on a given x- value or y-value, please show the substitution in your
regression equation accompanied by the answer.
Remember, x is the independent variable and y is the dependent variable.
These regression questions are nice, because they usually are not too difficult, HOWEVER, it is to very difficult to
find places to award partial credit if you make a mistake, so BE CAREFUL! Check to make sure you have entered
everything into the calculator appropriately and that you chose the equation they ask you to calculate.
Regression questions on previous exams have asked for: linear (LinReg - choice 4), exponential (Exp Reg - choice 0),
Logarithmic (LnREg - choice 9) or power (PwrReg - choice A) - but they can ask for anything that is listed in the
menu on your calculator.
NOTE: Please pay special attention to how they ask you to enter in the x-values if the variable describes time.
Sometimes they say the time is years since a certain year, for example:
The following values describe the number of I-pod sales since 2005, therefore 2005 would be year 0, 2006 would
be year 1, etc. You only do this if they ask you to!
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As shown in the table below, a person’s target heart rate during exercise changes as the person gets older.
Which value represents the linear correlation coefficient, rounded to the nearest thousandth, between a person’s age, in years, and that person’s target heart rate, in beats per minute?
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2. A population of single­celled organisms was grown in a Petri dish over a period of 16 hours. The number of organisms at a given time is recorded in the table below.
Determine the exponential regression equation model for these data, rounding all values to the nearest ten­thousandth. Using this equation, predict the number of single­celled organisms, to the nearest whole number
at the end of the 18th hour.
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3. The table below shows the results of an experiment that relates
the height at which a ball is dropped, x, to the height of its first
bounce, y.
Find x, the mean of drop heights.
Find y, the mean of bounce heights.
Find the linear regression line that best fits the data.
Show that (x, y) is a point on the regression line.
Drop Height (x) cm
Bounce Height (y) cm
100
26
90
23
80
21
70
18
60
16
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4. Based on a series of tests, here are the distances, in feet, it takes to stop a certain
car in minimum time under emergency conditions.
Speed (mph)
10
20
30
40
50
60
70
Distance 19
(feet)
42
73
116
173
248
343
a. Find the quadratic regression equation, round the coefficients to the nearest
thousandth.
b. Using the equation, how many feet would be needed for this car to stop if it
were traveling at a rate of 55mph?
c. Using this equation, if it took the car approximately 300 feet to stop, how fast
was the car traveling?
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5. 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
New Cases
2000
457
2001
369
2002
353
If x = 1 represents the year 2000, and y represents the number of new cases, find
the equation of best fit using a power regression, rounding all values to the nearest
thousandth.
Using this equation, find the estimated number of new cases, to the nearest whole
number, for the year 2007.
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6. A cup of soup is left on a countertop to cool. The table below gives the temperatures, in degrees Fahrenheit, of the soup recorded over a 10­minute period.
Write an exponential regression equation for the data, rounding all values to the nearest thousandth.
Using the equation, predict the temperature after 5 minutes.
To the nearest tenth of a minute, when would you expect the soup to reach 152 degrees. 8
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