1. Chapter 9 Estimation: Additional Topics
Statistics for
Business and Economics 6th Edition
Chapter 9
Estimation: Additional Topics

2. Chapter Goals After completing this chapter, you should be able to:
Form confidence intervals for the mean difference from dependent samples
Form confidence intervals for the difference between two independent population means (standard deviations known or unknown)
Compute confidence interval limits for the difference between two independent population proportions
Create confidence intervals for a population variance
Find chi-square values from the chi-square distribution table
Determine the required sample size to estimate a mean or proportion within a specified margin of error

3. Dependent Samples Tests Means of 2 Related Populations di = xi - yi
Paired or matched samples
Repeated measures (before/after)
Use difference between paired values:
Eliminates Variation Among Subjects
Assumptions:
Both Populations Are Normally Distributed
Dependent samples
di = xi - yi

4. Mean Difference di = xi - yi
The ith paired difference is di , where
di = xi - yi
The point estimate for the population mean paired difference is d :
The sample standard deviation is:
n is the number of matched pairs in the sample

5. Confidence Interval for Mean Difference
The confidence interval for difference between population means, μd , is
Where
n = the sample size
(number of matched pairs in the paired sample)

6. Paired Samples Example
Six people sign up for a weight loss program. You collect the following data: 
Weight:
Person Before (x) After (y) Difference, di 42 di
d = n
= 7.0

7. Paired Samples Example
For a 95% confidence level, the appropriate t value is tn-1,/2 = t5,.025 = 2.571
The 95% confidence interval for the difference between means, μd , is
Since this interval contains zero, we cannot be 95% confident, given this limited data, that the weight loss program helps people lose weight

8. Difference Between Two Means
Goal: Form a confidence interval for the difference between two population means, μx – μy
Population means, independent samples
Different data sources
Unrelated
Independent
Sample selected from one population has no effect on the sample selected from the other population
The point estimate is the difference between the two sample means:
x – y

9. σx2 and σy2 Unknown, Assumed Equal
Assumptions:
Samples are randomly and
independently drawn
Populations are normally
distributed
Population variances are
unknown but assumed equal
Population means, independent samples
σx2 and σy2 known
σx2 and σy2 unknown *
σx2 and σy2 assumed equal
σx2 and σy2 assumed unequal

10. σx2 and σy2 Unknown, Assumed Equal
The population variances
are assumed equal, so use
the two sample standard
deviations and pool them to
estimate σ
use a t value with
(nx + ny – 2) degrees of
freedom
Population means, independent samples
σx2 and σy2 known
σx2 and σy2 unknown *
σx2 and σy2 assumed equal
σx2 and σy2 assumed unequal

11. Confidence Interval, σx2 and σy2 Unknown, Equal*
σx2 and σy2 assumed equal
The confidence interval for
μ1 – μ2 is:
σx2 and σy2 assumed unequal

12. Pooled Variance Example
You are testing two computer processors for speed. Form a confidence interval for the difference in CPU speed. You collect the following speed data (in Mhz): CPUx CPUy Number Tested Sample mean Sample std dev 74 56
Assume both populations are
normal with equal variances, and use 95% confidence

13. Calculating the Pooled Variance
The pooled variance is:
The t value for a 95% confidence interval is:

14. Calculating the Confidence Limits
The 95% confidence interval is
We are 95% confident that the mean difference in CPU speed is between and Mhz.

15. Two Population Proportions
Goal: Form a confidence interval for the difference between two population proportions, Px – Py
Population proportions
Assumptions:
Both sample sizes are large (generally at least 40 observations in each sample)
The point estimate for the difference is
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

16. Two Population Proportions
(continued)
The random variable
is approximately normally distributed
Population proportions
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

17. Confidence Interval for Two Population Proportions
The confidence limits for
Px – Py are:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

18. Example: Two Population Proportions
Form a 90% confidence interval for the difference between the proportion of men and the proportion of women who have college degrees.
In a random sample, 26 of 50 men and 28 of 40 women had an earned college degree
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

19. Example: Two Population Proportions
(continued)
Men:
Women:
For 90% confidence, Z/2 = 1.645
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

20. Example: Two Population Proportions
(continued)
The confidence limits are:
so the confidence interval is
< Px – Py <
Since this interval does not contain zero we are 90% confident that the two proportions are not equal
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

21. Margin of Error
The required sample size can be found to reach a desired margin of error (ME) with a specified level of confidence (1 - )
The margin of error is also called sampling error
the amount of imprecision in the estimate of the population parameter
the amount added and subtracted to the point estimate to form the confidence interval
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

22. Sample Size Determination
Determining
Sample Size
For the
Mean
Margin of Error (sampling error)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

23. Sample Size Determination
(continued)
Determining
Sample Size
For the
Mean
Now solve for n to get
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

24. Sample Size Determination
(continued)
To determine the required sample size for the mean, you must know:
The desired level of confidence (1 - ), which determines the z/2 value
The acceptable margin of error (sampling error), ME
The standard deviation, σ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

25. Required Sample Size Example
If  = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence?
So the required sample size is n = 220
(Always round up)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

26. Sample Size Determination
Determining
Sample Size
For the
Proportion
Margin of Error (sampling error)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

27. Sample Size Determination
(continued)
Determining
Sample Size
For the
Proportion
cannot be larger than 0.25, when = 0.5
Substitute 0.25 for
and solve for n to get
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

28. Sample Size Determination
(continued)
The sample and population proportions, and P, are generally not known (since no sample has been taken yet)
P(1 – P) = 0.25 generates the largest possible margin of error (so guarantees that the resulting sample size will meet the desired level of confidence)
To determine the required sample size for the proportion, you must know:
The desired level of confidence (1 - ), which determines the critical z/2 value
The acceptable sampling error (margin of error), ME
Estimate P(1 – P) = 0.25
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

29. Required Sample Size Example
How large a sample would be necessary to estimate the true proportion defective in a large population within ±3%, with 95% confidence?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

30. Required Sample Size Example
(continued)
Solution:
For 95% confidence, use z0.025 = 1.96
ME = 0.03
Estimate P(1 – P) = 0.25
So use n = 1068
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.