1. Chapter 9 Estimation: Additional Topics
Statistics for
Business and Economics 6th Edition
Chapter 9
Estimation: Additional Topics
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

3. Estimation: Additional Topics
Chapter Topics
Population Means, Dependent Samples
Population Means, Independent Samples
Population Proportions
Population Variance
Examples:
Same group before vs. after treatment
Group 1 vs. independent Group 2
Proportion 1 vs. Proportion 2
Variance of a normal distribution
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

4. 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

5. Mean Difference The ith paired difference is di , where di = xi - yi
Dependent samples
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

6. Confidence Interval for Mean Difference
The confidence interval for difference between population means, μd , is
Dependent samples
Where
n = the sample size
(number of matched pairs in the paired sample)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

7. Confidence Interval for Mean Difference
(continued)
The margin of error is
tn-1,/2 is the value from the Student’s t distribution with (n – 1) degrees of freedom for which
Dependent samples
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

8. 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

9. Paired Samples Example
(continued)
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

10. 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

11. Difference Between Two Means
(continued)
Population means, independent samples
σx2 and σy2 known
Confidence interval uses z/2
σx2 and σy2 unknown
σx2 and σy2 assumed equal
Confidence interval uses a value from the Student’s t distribution
σx2 and σy2 assumed unequal
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

12. Population means, independent samples
σx2 and σy2 Known
Population means, independent samples
Assumptions:
Samples are randomly and
independently drawn
both population distributions
are normal
Population variances are
known *
σx2 and σy2 known
σx2 and σy2 unknown
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

13. Population means, independent samples
σx2 and σy2 Known
(continued)
When σx and σy are known and both populations are normal, the variance of X – Y is
Population means, independent samples *
σx2 and σy2 known
…and the random variable
has a standard normal distribution
σx2 and σy2 unknown
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

14. Confidence Interval, σx2 and σy2 Known
Population means, independent samples *
σx2 and σy2 known
The confidence interval for
μx – μy is:
σx2 and σy2 unknown
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

15. 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

16. σx2 and σy2 Unknown, Assumed Equal
(continued)
Forming interval estimates:
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

17. σx2 and σy2 Unknown, Assumed Equal
(continued)
Population means, independent samples
The pooled variance is
σx2 and σy2 known
σx2 and σy2 unknown *
σx2 and σy2 assumed equal
σx2 and σy2 assumed unequal
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

18. Confidence Interval, σx2 and σy2 Unknown, Equal*
σx2 and σy2 assumed equal
The confidence interval for
μ1 – μ2 is:
σx2 and σy2 assumed unequal
Where
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

19. 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
Assume both populations are
normal with equal variances, and use 95% confidence
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

20. Calculating the Pooled Variance
The pooled variance is:
The t value for a 95% confidence interval is:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

21. Calculating the Confidence Limits
The 95% confidence interval is
We are 95% confident that the mean difference in CPU speed is between and Mhz.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

22. σx2 and σy2 Unknown, Assumed Unequal
Assumptions:
Samples are randomly and
independently drawn
Populations are normally
distributed
Population variances are
unknown and assumed
unequal
Population means, independent samples
σx2 and σy2 known
σx2 and σy2 unknown
σx2 and σy2 assumed equal *
σx2 and σy2 assumed unequal
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

23. σx2 and σy2 Unknown, Assumed Unequal
(continued)
Forming interval estimates:
The population variances are
assumed unequal, so a pooled
variance is not appropriate
use a t value with  degrees
of freedom, where
Population means, independent samples
σx2 and σy2 known
σx2 and σy2 unknown
σx2 and σy2 assumed equal *
σx2 and σy2 assumed unequal
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

24. Confidence Interval, σx2 and σy2 Unknown, Unequal
σx2 and σy2 assumed equal
The confidence interval for
μ1 – μ2 is: *
σx2 and σy2 assumed unequal
Where
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

25. 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.

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

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

28. 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.

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

30. 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.

31. Confidence Intervals for the Population Variance
Goal: Form a confidence interval
for the population variance, σ2
The confidence interval is based on the sample variance, s2
Assumed: the population is normally distributed
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

32. Confidence Intervals for the Population Variance
(continued)
Population
Variance
The random variable
follows a chi-square distribution with (n – 1) degrees of freedom
The chi-square value denotes the number for which
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

33. Confidence Intervals for the Population Variance
(continued)
Population
Variance
The (1 - )% confidence interval for the population variance is
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

34. Example
You are testing the speed of a computer processor. You collect the following data (in Mhz):
CPUx Sample size
Sample mean
Sample std dev
Assume the population is normal.
Determine the 95% confidence interval for σx2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

35. Finding the Chi-square Values
n = 17 so the chi-square distribution has (n – 1) = 16 degrees of freedom
 = 0.05, so use the the chi-square values with area in each tail:
probability α/2 = .025
probability α/2 = .025
216
216
= 6.91
216
= 28.85
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

36. Calculating the Confidence Limits
The 95% confidence interval is
Converting to standard deviation, we are 95% confident that the population standard deviation of CPU speed is between 55.1 and Mhz
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

37. Sample PHStat Output
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

38. Sample PHStat Output Input Output (continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

39. Sample Size Determination
Determining
Sample Size
For the
Mean
For the
Proportion
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

40. 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.

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

42. 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.

43. 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.

44. 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.

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

46. 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.

47. 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.

48. 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.

49. 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.

50. PHStat Sample Size Options
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

51. Chapter Summary Compared two dependent samples (paired samples)
Formed confidence intervals for the paired difference
Compared two independent samples
Formed confidence intervals for the difference between two means, population variance known, using z
Formed confidence intervals for the differences between two means, population variance unknown, using t
Formed confidence intervals for the differences between two population proportions
Formed confidence intervals for the population variance using the chi-square distribution
Determined required sample size to meet confidence and margin of error requirements
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.