1. Comparing k Populations
Means – One way Analysis of Variance (ANOVA)

2. The F test – for comparing k means
Situation
We have k normal populations
Let mi and s denote the mean and standard deviation of population i.
i = 1, 2, 3, … k.
Note: we assume that the standard deviation for each population is the same.
s1 = s2 = … = sk = s

3. We want to test
against

4. A convenient method for displaying the calculations for the F-test
The ANOVA Table
A convenient method for displaying the calculations for the F-test

5. Anova Table Mean Square F-ratio Between k - 1 SSBetween MSBetween
Source
d.f.
Sum of Squares
Mean Square
F-ratio
Between
k - 1
SSBetween
MSBetween
MSB /MSW
Within
N - k
SSWithin
MSWithin
Total
N - 1
SSTotal

6. To Compute F (and the ANOVA table entries):1) 2) 3) 4) 5)

7. Then 1) 2) 3) 4)

8. The c2 test for independence

9. Situation We have two categorical variables R and C.
The number of categories of R is r.
The number of categories of C is c.
We observe n subjects from the population and count xij = the number of subjects for which R = I and C = j.
R = rows, C = columns

10. Example
Both Systolic Blood pressure (C) and Serum Cholesterol (R) were meansured for a sample of n = 1237 subjects.
The categories for Blood Pressure are:
<
The categories for Cholesterol are:
<

11. Table: two-way frequency

12. The c2 test for independence
Define
= Expected frequency in the (i,j) th cell in the case of independence.

13. Justification - for Eij = (RiCj)/n in the case of independence
Let pij = P[R = i, C = j] = P[R = i] P[C = j] = rigj in the case of independence
= Expected frequency in the (i,j) th cell in the case of independence.

14. H0: R and C are independent
Then to test
H0: R and C are independent
against
HA: R and C are not independent
Use test statistic
Eij= Expected frequency in the (i,j) th cell in the case of independence.
xij= observed frequency in the (i,j) th cell

15. Sampling distribution of test statistic when H0 is true
- c2 distribution with degrees of freedom n = (r - 1)(c - 1)
Critical and Acceptance Region
Reject H0 if :
Accept H0 if :

17. Standardized residuals
Test statistic
degrees of freedom n = (r - 1)(c - 1) = 9
Reject H0 using a = 0.05

18. Hypothesis testing and Estimation
Linear Regression
Hypothesis testing and Estimation

19. Fitting the best straight line to “linear” data
The Least Squares Line
Fitting the best straight line
to “linear” data

20. The equation for the least squares line
Let

21. Computing Formulae:

22. Then the slope of the least squares line can be shown to be:

23. and the intercept of the least squares line can be shown to be:

24. The residual sum of Squares
Computing formula

25. Estimating s, the standard deviation in the regression model :
Computing formula
This estimate of s is said to be based on n – 2 degrees of freedom

26. Sampling distributions of the estimators

27. The sampling distribution slope of the least squares line :
It can be shown that b has a normal distribution with mean and standard deviation

28. The sampling distribution intercept of the least squares line :
It can be shown that a has a normal distribution with mean and standard deviation

29. Estimating s, the standard deviation in the regression model :
Computing formula
This estimate of s is said to be based on n – 2 degrees of freedom

30. (1 – a)100% Confidence Limits for slope b :
ta/2 critical value for the t-distribution with n – 2 degrees of freedom

31. (1 – a)100% Confidence Limits for intercept a :
ta/2 critical value for the t-distribution with n – 2 degrees of freedom

32. Example
In this example we are studying building fires in a city and interested in the relationship between:
X = the distance of the closest fire hall and the building that puts out the alarm
and
Y = cost of the damage (1000$)
The data was collected on n = 15 fires.

33. The Data

34. Scatter Plot

35. Computations

36. Computations Continued

37. Computations Continued

38. Computations Continued

39. Least Squares Line
y=4.92x+10.28

40. 95% Confidence Limits for slope b :
4.07 to 5.77
t.025 = critical value for the t-distribution with 13 degrees of freedom

41. 95% Confidence Limits for intercept a :
7.21 to 13.35
t.025 = critical value for the t-distribution with 13 degrees of freedom