1. Chapter 4
Correlation

2. Suppose we found the age and weight of a sample of 10 adults.
Create a scatterplot of the data below.
Is there any relationship between the age and weight of these adults?
Age 24 30 41 28 50 46 49 35 20 39 Wt
256
124
320
185
158
129
103
196
110
130

3. Create a scatterplot of the data below.
Suppose we found the height and weight of a sample of 10 adults.
Create a scatterplot of the data below.
Is there any relationship between the height and weight of these adults?
Is it positive or negative? Weak or strong? Ht 74 65 77 72 68 60 62 73 61 64 Wt
256
124
320
185
158
129
103
196
110
130

4. The farther away from a straight line – the weaker the relationship
The closer the points in a scatterplot are to a straight line - the stronger the relationship.
The farther away from a straight line – the weaker the relationship

5. Identify as having a positive association, a negative association, or no association.+
Heights of mothers & heights of their adult daughters -
Age of a car in years and its current value +
Weight of a person and calories consumed
Height of a person and the person’s birth month NO
Number of hours spent in safety training and the number of accidents that occur -

6. Correlation Coefficient (r)-
A quantitative assessment of the strength & direction of the linear relationship between bivariate, quantitative data
Use the following sentence to describe r:
There is a direction (positive/negative), strength (strong/moderate/weak), linear association between x and y.

7. Correlation Coefficient (r)-
Pearson’s sample correlation is used most
parameter - r (rho)
statistic - r

8. Calculate r. Interpret r in context.
Speed Limit (mph) 55 50 45 40 30 20
Avg. # of accidents (weekly) 28 25 21 17 11 6
Calculate r. Interpret r in context.
There is a strong, positive, linear relationship between speed limit and average number of accidents per week.

9. Properties of r (correlation coefficient)
legitimate values of r is [-1,1]
Strong correlation No
Correlation
Moderate Correlation
Weak correlation

10. The correlations are the same.
value of r does not depend on the unit of measurement for either variable
x (in mm) y
Find r.
Change to cm & find r.
The correlations are the same.

11. value of r does not depend on which of the two variables is labeled xy
Switch x & y & find r.
The correlations are the same.

12. value of r is non-resistantx y
Find r.
Outliers affect the correlation coefficient

13. r = 0, but has a definite relationship!
value of r is a measure of the extent to which x & y are linearly related
Find the correlation for these points: x Y
Sketch the scatterplot
r = 0, but has a definite relationship!

14. Association vs. Causation
In a famous example of a correlation study, the following results were obtained.
Year Number of Methodist Cuban Rum Imported
Ministers in New England to Boston (in barrels)
,643
,265
,071
,547
,008
,885
,559
,024
,185
,434
,238
,705

15. Minister data:
r = .9999
So does an increase in ministers cause an increase in consumption of rum?

16. Correlation does not imply causation

17. LSRL
Section 4.2

18. Bivariate data
x – variable: is the independent or explanatory variable
y- variable: is the dependent or response variable
Use x to predict y

19. Be sure to put the hat on the y
- (y-hat) means the predicted y
b – is the slope
it is the approximate amount by which y increases when x increases by 1 unit
a – is the y-intercept
it is the approximate height of the line when x = 0
in some situations, the y-intercept has no meaning
Be sure to put the hat on the y

20. Least Squares Regression Line LSRL
The line that gives the best fit to the data set
The line that minimizes the sum of the squares of the deviations from the line

21. (0,0)
(3,10)
(6,2)
y =.5(6) + 4 = 7
2 – 7 = -5
4.5
y =.5(0) + 4 = 4
0 – 4 = -4 -5
y =.5(3) + 4 = 5.5
10 – 5.5 = 4.5 -4
(0,0)
Sum of the squares = 61.25

22. What is the sum of the deviations from the line?
Will it always be zero?
Use a calculator to find the line of best fit
(0,0)
(3,10)
(6,2) 6
Find y - y -3
The line that minimizes the sum of the squares of the deviations from the line is the LSRL. -3
Sum of the squares = 54

23. Interpretations Slope:
For each unit increase in x, there is an approximate increase/decrease of b in y.
Correlation coefficient:
There is a direction, strength, linear association between x and y.

24. The ages (in months) and heights (in inches) of seven children are given.x y
Find the LSRL.
Interpret the slope and correlation coefficient in the context of the problem.

25. Correlation coefficient:
There is a strong, positive, linear association between the age and height of children.
Slope:
For an increase in age of one month, there is an approximate increase of .34 inches in heights of children.

26. Predict the height of a child who is 4.5 years old.
The ages (in months) and heights (in inches) of seven children are given. x y
Predict the height of a child who is 4.5 years old.
Predict the height of someone who is 20 years old.
Graph, find lsrl, also examine mean of x & y

27. Extrapolation
The LSRL should not be used to predict y for values of x outside the data set.
It is unknown whether the pattern observed in the scatterplot continues outside this range.

28. For these data, this is the best equation to predict y from x.
The ages (in months) and heights (in inches) of seven children are given.
The LSRL is
Can this equation be used to estimate the age of a child who is 50 inches tall?
Calculate: LinReg L2,L1
For these data, this is the best equation to predict y from x.
Do you get the same LSRL?
However, statisticians will always use this equation to predict x from y

29. Will this point always be on the LSRL?
The ages (in months) and heights (in inches) of seven children are given. x y
Calculate x & y.
Plot the point (x, y) on the LSRL.
Graph, find lsrl, also examine mean of x & y
Will this point always be on the LSRL?

30. The correlation coefficient and the LSRL are both non-resistant measures.

31. Formulas – on chart

32. The following statistics are found for the variables posted speed limit and the average number of accidents.
Find the LSRL & predict the number of accidents for a posted speed limit of 50 mph.