1. Chapter 11 Comparisons Involving Proportions and a Test of Independence
Inference about the Difference Between the
Proportions of Two Populations
A Hypothesis Test for Proportions of a
Multinomial Population
Test of Independence: Contingency Tables
Ho: p1 - p2 = 0
Ha: p1 - p2 = 0

2. Inferences About the Difference between the Proportions of Two Populations
Sampling Distribution of
Interval Estimation of p1 - p2
Hypothesis Tests about p1 - p2

3. Sampling Distribution of
Expected Value
Standard Deviation
where: n1 = size of sample taken from population 1
n2 = size of sample taken from population 2

4. Sampling Distribution of
Distribution Form
If the sample sizes are large (n1p1, n1(1 - p1), n2p2,
and n2(1 - p2) are all greater than or equal to 5), the
sampling distribution of can be approximated
by a normal probability distribution.

5. Sampling Distribution of
p1 – p2

6. Interval Estimation of p1 - p2
(Confidence) Interval Estimate
Point Estimator of

7. Example: MRA
MRA (Market Research Associates) is conducting research to evaluate the effectiveness of a client’s new advertising campaign. Before the new campaign began, a telephone survey of 150 households in the test market area showed 60 households “aware” of the client’s product. The new campaign has been initiated with TV and newspaper advertisements running for three weeks.

8. Example: MRA
A survey conducted immediately after the new campaign showed 120 of 250 households “aware” of the client’s product.
Does the data support the position that the advertising campaign has provided an increased awareness of the client’s product?

9. Example: MRA
Point Estimator of the Difference Between the Proportions of Two Populations is
p1 = proportion of the population of households
“aware” of the product after the new campaign
p2 = proportion of the population of households
“aware” of the product before the new campaign
= sample proportion of households “aware” of the
product after the new campaign
product before the new campaign

10. Example: MRA Interval Estimate of p1 - p2: Large-Sample Case
For = .05, z.025 = _______:
(.0510)
-.02 to +.18

11. Example: MRA Interval Estimate of p1 - p2: Large-Sample Case
Conclusion
At a 95% confidence level, the interval estimate of the difference between the proportion of households aware of the client’s product before and after the new advertising campaign is ___ to _____.

12. Hypothesis Tests about p1 - p2
Hypotheses
H0: p1 - p2 < 0
Ha: p1 - p2 > 0
Test statistic
where takes the _______ part of the value in H0.

13. Hypothesis Tests about p1 - p2
Point Estimator of where p1 = p2
where:

14. Example: MRA Hypothesis Tests about p1 - p2
Can we conclude, using a .05 level of significance, that the proportion of households aware of the client’s product increased after the new advertising campaign?

15. Example: MRA Hypothesis Tests about p1 - p2 Hypotheses
H0: p1 - p2 < 0
Ha: p1 - p2 > 0
p1 = proportion of the population of households
“aware” of the product ______ the new campaign
p2 = proportion of the population of households

16. Example: MRA Hypothesis Tests about p1 - p2
Rejection Rule Reject H0 if z > 1.645
Test Statistic

17. Example: MRA Hypothesis Tests about p1 - p2 Conclusion
z = ______ < Do not reject H0. We cannot conclude, with at least 95 % confidence, that the proportion of households aware of the client’s product increased after the new advertising campaign.

18. Hypothesis (Goodness of Fit) Test for Proportions of a Multinomial Population
1. Set up the null and alternative hypotheses.
2. Select a random sample and record the observed
frequency, fi , for each of the k categories.
3. Assuming H0 is true, compute the expected frequency, ei , in each category by multiplying the category probability by the sample size.

19. Hypothesis (Goodness of Fit) Test for Proportions of a Multinomial Population
4. Compute the value of the test statistic.
5. Reject H0 if (where  is the significance level and there are k - 1 degrees of freedom).

20. Example: Finger Lakes Homes (A)
Multinomial Distribution Goodness of Fit Test
Finger Lakes Homes manufactures four models of
prefabricated homes, a two-story colonial, a ranch, a
split-level, and an A-frame. To help in production
planning, management would like to determine if
previous customer purchases indicate that there is a
preference in the style selected.

21. Example: Finger Lakes Homes (A)
Multinomial Distribution Goodness of Fit Test
The number of homes sold of each model for ______
sales over the past two years is shown below.
Model Colonial Ranch Split-Level A-Frame
# Sold

22. Example: Finger Lakes Homes (A)
Multinomial Distribution Goodness of Fit Test
Let:
pC = population proportion that purchase a colonial
pR = population proportion that purchase a ranch
pS = population proportion that purchase a split-level
pA = population proportion that purchase an A-frame

23. Example: Finger Lakes Homes (A)
Multinomial Distribution Goodness of Fit Test
Hypotheses
H0: pC = pR = pS = pA = .25
Ha: The population proportions are not pC = .25,
pR = .25, pS = .25, and pA = .25

24. Example: Finger Lakes Homes (A)
Multinomial Distribution Goodness of Fit Test
Rejection Rule
With  = .05 and
k - 1 = = 3 degrees of
freedom
Do Not Reject H0
Reject H0 2
7.815

25. Example: Finger Lakes Homes (A)
Multinomial Distribution Goodness of Fit Test
Expected Frequencies
e1 = .25(100) = e2 = .25(100) = 25
e3 = .25(100) = e4 = .25(100) = 25
Test Statistic =
= 10

26. Example: Finger Lakes Homes (A)
Multinomial Distribution Goodness of Fit Test
Conclusion
c2 = 10 > We reject the assumption there is no home style preference, at the .05 level of significance.

27. Test of Independence: Contingency Tables
1. Set up the null and alternative hypotheses.
2. Select a random sample and record the observed
frequency, fij , for each cell of the contingency table.
3. Compute the expected frequency, eij , for each cell.

28. Test of Independence: Contingency Tables
4. Compute the test statistic.
5. Reject H0 if (where  is the significance level and with n rows and m columns there are
(n - 1)(m - 1) degrees of freedom).

29. Example: Finger Lakes Homes (B)
Contingency Table (Independence) Test
Each home sold can be classified according to price and to style. Finger Lakes Homes’ manager would like to determine if the price of the home and the style of the home are independent variables.

30. Example: Finger Lakes Homes (B)
Contingency Table (Independence) Test
The number of homes sold for each model and price for the past two years is shown below. For convenience, the price of the home is listed as either $99,000 or less or more than $99,000.
Price Colonial Ranch Split-Level A-Frame
< $99,
> $99,

31. Example: Finger Lakes Homes (B)
Contingency Table (Independence) Test
Hypotheses
H0: Price of the home is independent of the style of the home that is purchased
Ha: Price of the home is not independent of the
style of the home that is purchased

32. Example: Finger Lakes Homes (B)
Contingency Table (Independence) Test
Expected Frequencies
Price Colonial Ranch Split-Level A-Frame Total
< $99K
> $99K
Total

33. Example: Finger Lakes Homes (B)
Contingency Table (Independence) Test
Rejection Rule
With  = .05 and (2 - 1)(4 - 1) = 3 d.f.,
Reject H0 if 2 > 7.81

34. Example: Finger Lakes Homes (B)
Contingency Table (Independence) Test
Test Statistic
= = ______

35. Example: Finger Lakes Homes (B)
Contingency Table (Independence) Test
Conclusion
2 = ____ > 7.81, so we reject H0, the assumption that the price of the home is independent of the style of the home that is purchased.

36. End of Chapter 11