Download Math 227 Ch 4 notes KEY

Chapter 4 Describing the Relation between Two Variables
4.1 Scatter Diagrams and Correlation
The RESPONSE is the variable whose value can be explained by the value of the EXPLANATORY or
PREDICTOR VARIABLE. ‘y’ depends on ‘x’
A SCATTER DIAGRAM is a graph that shows the relationship between two quantitative variables
measured on the same individual. Each individual in the data set is represented by a point in the scatter
diagram.
I. Scatter plot
(x)
# hour
of sleep
6
8
10
2
(y)
performance
3
5
4
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II. Linear Correlation coefficient (r)
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The linear correlation coefficient or Pearson product moment correlation coefficient is a measure of the
strength and direction of the linear relation between two quantitative variables. The Greek letter ρ (rho)
represents the population correlation coefficient, and r represents the sample correlation coefficient.
We present only the formula for the sample correlation coefficient.
Sample Linear Correlation Coefficient
 xi  x   yi  y 
  s   s 
 x  y 
r
n 1
where x is the sample mean of the explanatory variable
sx is the sample standard deviation of the explanatory variable
y is the sample mean of the response variable
sy is the sample standard deviation of the response variable
n is the number of individuals in the sample
Properties of the Linear Correlation Coefficient
1. The linear correlation coefficient is always between –1 and 1, inclusive. That is, –1 ≤ r ≤ 1.
2. If r = + 1, then a perfect positive linear relation exists between the two variables.
3. If r = –1, then a perfect negative linear relation exists between the two variables.
4. The closer r is to +1, the stronger is the evidence of positive association between the two
variables.
5. The closer r is to –1, the stronger is the evidence of negative association between the two
variables.
6. If r is close to 0, then little or no evidence exists of a linear relation between the two variables.
So r close to 0 does not imply no relation, just no linear relation.
7. The linear correlation coefficient is a unitless measure of association. So the unit of measure for
x and y plays no role in the interpretation of r.
8. The correlation coefficient is not resistant. Therefore, an observation that does not follow the
overall pattern of the data could affect the value of the linear correlation coefficient.
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EXAMPLE Determining the Linear Correlation Coefficient
Determine the linear correlation coefficient of the drilling data.
STEP 1: First make a scatterplot (using stat crunch)
to see if there is a linear relationship! Put data in column 1 & 2
Step 2: STAT-SUMMARY STATISTICS-CORRELATION-SELECT COLUMNSCLICK 1ST VAR & CLICK 2ND VAR WHILE HOLDING CTRL KEY-COMPUTE
r = 0.7773
Compare to diagram:
Testing for a Linear Relation
• Step 1 Determine the absolute value of the correlation coefficient
• Step 2 Find the critical value in Table II from Appendix A for the given sample size
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• Step 3 If the absolute value of the correlation coefficient is greater than the critical
value, we say a linear relation exists between the two variables. Otherwise, no linear
relation exists.
EXAMPLE Does a Linear Relation Exist?
R=0.7773 There is a strong positive correlation between depth at which drilling begins and time
to drill 5 feet.
Using the table n = 12 CV 0.576
Since .7773 > 0.576 then a linear relationship exists.
4.2 Least-Squares Regression
EXAMPLE Finding an Equation that Describes Linearly Related Data
The following data shows the number of doctor visits in a year (x) with the corresponding
days sick from work (y) for six patients at a clinic.
(a) Graph the equation on the scatter diagram.
(b) Find a linear equation that relates x (the explanatory variable) and y (the response
variable) by selecting two points on the line of best fit and finding the equation of the
line containing the points.
Use (5, 3) and (1, 6)
63
3
  =-0.75
Slope =
1 5
4
Equation: y  y1  m( x  x1 )
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y  6  0.75( x  1)
y  6  0.75 x  0.75
y  0.75 x  6.75 or
Days sick = -0.75(doctor visits) + 6.75
S = - 0.75 V + 6.75
(c) Use the equation to predict the number of sick days if you visit the doctor 3 times a year.
S = - 0.75 V + 6.75
S = - 0.75 (3) + 6.75 = - 2.25 + 6.75 = 4.5 days
If you visit the doctor three times a year, you can expect to miss 4.5 days of work on average.
d) Does going to the doctor cause you to miss less days?
No. association ≠ causation
It may cause you to miss less days.
e) Use the equation to predict the number of sick days if you visit the doctor 10 times a
year.
You cannot predict beyond the scope of the data!!!! This is called extrapolation.
The difference between the observed value of y and the predicted value of y is the error, or residual.
Using the line from the last example, and the predicted value at x = 3:
residual = observed y – predicted y = 5.2 – 4.5 = 0.7 days (under predicted)
Least-Squares Regression Criterion
If there is positive / negative correlation between x and y, find the best fitted line for the data.
The least-squares regression line is the line that minimizes the sum of the squared errors (or residuals).
This line minimizes the sum of the squared vertical distance between the observed values of y and those
predicted by the line ŷ , (“y-hat”). We represent this as “minimize Σ residuals2 ” (minimizes the sum of
the squared errors).
The Least-Squares Regression Line
The equation of the least-squares regression line is given by yˆ  b1 x  b0
where b1  r
sy
sx
is the slope of the least-squares regression line
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and b0  y  b1 x is the y-intercept of the least-squares regression line
The Least-Squares Regression Line
Note: x is the sample mean and sx is the sample standard deviation of the explanatory variable x ; y
is the sample mean and sy is the sample standard deviation of the response variable y.
EXAMPLE Finding the Least-squares Regression Line
Using the drilling data and computer technology
(a) Draw the least-squares regression line on the scatter diagram of the data
to verify a linear relationship.
From before r = .773
(b) Find the least-squares regression line.
1.Input data into statcrunch
2. STAT-REGRESSION-STIMPLE LINEAR-SELECT VAR’S-COMPUTE
3. equation: yˆ  5.5273  0.0116 x
(c) Predict the drilling time if drilling starts at 130 feet.
ŷ  5.5273  0.0116(130)  7.035 minutes
(d) Is the observed drilling time at 130 feet above, or below, average.
Observered = 6.93 minutes which was below predicted average.
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e) Interpretation of Slope:
0.0116 min
1 foot
For each additional one foot of drilling, the time to drill 5 feet increases by 0.0116 minutes on
average.
Interpretation of the y-Intercept: 5.5273 ≈5.5 feet (occurs when x = 0)
When drilling begins at 0 feet (the surface), the time to drill 5 feet is 5.5 minutes.
Caution: If the least-squares regression line is used to make predictions based on values of the
explanatory variable that are much larger or much smaller than the observed values, we say the
researcher is working outside the scope of the model. Never use a least-squares regression line to make
predictions outside the scope of the model because we can’t be sure the linear relation continues to
exist.
Predictions When There is No Linear Relation: No predictions should be made!
When the correlation coefficient indicates no linear relation between the explanatory and response
variables, and the scatter diagram indicates no relation at all between the variables, then we use the
mean value of the response variables, then we use the mean value of the response variable as the
predicted value so that ŷ  y
Summary
1. Use StatCrunch to plot a scatter plot
2. Use StatCrunch to calculate r
3. Determine whether there is a positive/negative linear correlation between X and Y.
4. If there is a linear correlation between X and Y, use StatCrunch to find the least squares regression
line. Otherwise, do not find the least squares regression line. And stop!
5. When a value is assigned to X  if there is a correlation between X and Y, use the least squares
regression line to find the best predicted Y.
6. When a value is assigned to X  if there is no correlation between X and Y, use StatCrunch to find
y and the best predicted Y is y for any X.
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