1. Hypothesis Testing and Confidence Intervals
Two Population Means
Hypothesis Testing and Confidence Intervals
For Differences in Proportions

2. SITUATION: 2 Populations
Yes = 1
No = 0
Men - Republican?
Women - Republican?
TV in Cuba - Good?
TV in China - Good?
HS >$100K?
College > $100K?
Trad. Course “A”?
Internet Course “A”?

3. Notation

4. Individual Responses Proportion of Responses
Each Individual Response is:
1 = YES or 0 = NO
Bernoulli Distribution:
Mean = p; Variance = pq (where q = (1-p))
The Average or Proportion of n responses (n large) by the Central Limit Theorem is:
Distributed approximately normal with mean = p
Variance = pq/n

5. Differences in Proportions
Proportion of “1’s” from Population 1 has:
a normal distribution (approximately)
true mean: p1 (which we don’t know)
true variance p1q1/n1
Proportion of “1’s” from Population 2 has:
true mean: p2 (which we don’t know)
true variance p2q2/n2

6. Distribution of the Difference in Proportions
True mean: p1 - p2
True variance p1q1/n1 + p2q2/n2
True standard deviation: SQRT(p1q1/n1 + p2q2/n2)
But since p1 and p2 are unknown, what should we use for the standard deviation in confidence intervals and hypothesis tests?
For all confidence intervals
and for all hypothesis tests
except H0: p1- p2 = 0
For hypothesis tests
of the form
H0: p1- p2 = 0

7. Hypothesis Tests and Confidence Intervals
Z-Statistics for Difference in Proportions
H0: p1- p2 = v
Where v ≠ 0
H0: p1- p2 = 0
Confidence Interval

8. Example
Midas wants to compare customer satisfaction between NY and LA operations. Can it conclude:
(1) A greater proportion in LA are satisfied?
(2) Customer satisfaction in LA exceeds NY by > 2%?
(3) Give a 95% confidence interval for difference in customer satisfaction.
Results --
350 out of 400 in LA were satisfied
160 out of 200 in NY were satisfied

9. (1) Is a greater proportion in LA?
H0: p1 - p2 = 0
HA: p1 - p2 > 0
Select α = .05
Reject H0 (Accept HA) if z > z.05 = 1.645
2.425 > 1.645; so it can be concluded that a greater proportion of Midas customers are satisfied customers in LA compared to New York.
This is a hypothesis test with v = 0. Use

10. (2) Is Customer Satisfaction more than 2% greater in LA?
H0: p1 - p2 = .02
HA: p1 - p2 > .02
Select α = .05.
Reject H0 (Accept HA) if z > z.05 = 1.645
1.679 > 1.645, so it can be concluded that customer satisfaction in LA exceeds that in New York by more than 2%.
This is a hypothesis test with v ≠ 0. Use

11. 95% Confidence Interval for the Difference in Proportions

12. Excel – Differences in Proportions
=COUNTA(A2:A401)
=COUNTA(B2:B201)
=F1+F2
=COUNTIF(A2:A401,“YES”)
=COUNTIF(B2:B201,“YES”)
=F5+F6
=F5/F1
=F6/F2
=F7/F3
=(F9-F10-0)/SQRT(F11*(1-F11)*(1/F1+1/F2))
=1-NORMSDIST(E14)
=(F9-F10-.02)/SQRT((F9*(1-F9)/F1)+(F10*(1-F10)/F2))
=1-NORMSDIST(E19)
=(F9-F10)-NORMSINV(.975)*SQRT((F9*(1-F9)/F1)+(F10*(1-F10)/F2))
=(F9-F10)+NORMSINV(.975)*SQRT((F9*(1-F9)/F1)+(F10*(1-F10)/F2))

13. Determining Sample Sizes
Usual Assumptions:
Sample Sizes equal n1 = n2 = n
Take the “worst case scenario” for the standard deviation -- when p1 = p2 = (unless you have reason to believe otherwise)
Use the “±” part of the confidence interval
How many people need to be surveyed in each city to estimate the difference in customer satisfaction to within ± 4%?

14. No Idea of the Proportions

15. In a Recent Survey 90% in LA and 75% in NY Were Satisfied

16. Review Confidence Intervals for difference in proportions
Hypothesis Tests for difference in proportions of the form p1 - p2 = 0
By hand and by Excel
Hypothesis Tests for difference in proportions of the form p1 - p2 = d
Confidence Intervals for difference in proportions
Estimating Sample Sizes
No idea of the proportions
Some idea of the proportions