1. Chapter 8 Estimation: Additional Topics
Statistics for
Business and Economics 8th Edition
Chapter 8
Estimation: Additional Topics
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Chapter Goals
After completing this chapter, you should be able to:
Form confidence intervals for the difference between two means 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
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3. Estimation: Additional Topics
Chapter Topics
Confidence Intervals
Population Means, Dependent Samples
Population Means, Independent Samples
Population Proportions
Examples:
Same group before vs. after treatment
Group 1 vs. independent Group 2
Proportion 1 vs. Proportion 2
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Dependent Samples
8.1
Confidence Interval Estimation of the Difference
Between Two Normal Population Means: Dependent Samples
Dependent samples
Tests Means of 2 Related Populations
Paired or matched samples
Repeated measures (before/after)
Use difference between paired values:
Eliminates Variation Among Subjects
Assumptions:
Both Populations Are Normally Distributed
di = xi - yi
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Mean Difference
The ith paired difference is di , where
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
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6. Confidence Interval for Mean Difference
The confidence interval for the difference between two population means, μd , is
Dependent samples
Where
n = the sample size
(number of matched pairs in the paired sample)
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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
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8. Paired Samples Example
Six people sign up for a weight loss program. You collect the following data:
Dependent samples 
Weight:
Person Before (x) After (y) Difference, di 42 di
d = n
= 7.0
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9. Paired Samples Example
(continued)
Dependent samples
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
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10. Difference Between Two Means: Independent Samples
8.2
Confidence Interval Estimation of the Difference Between Two Normal Population Means: Independent Samples
Population means, independent samples
Goal: Form a confidence interval for the difference between two population means, μx – μy
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11. Difference Between Two Means: Independent Samples
(continued)
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
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12. Difference Between Two Means: Independent Samples
(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
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13. * σx2 and σy2 Known Assumptions: Population means, independent samples
Samples are randomly and
independently drawn
both population distributions
are normal
Population variances are
known *
σx2 and σy2 known
σx2 and σy2 unknown
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14. 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
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15. 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
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16. 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
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17. 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
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18. 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
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19. 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
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20. 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
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21. Calculating the Pooled Variance
The pooled variance is:
The t value for a 95% confidence interval is:
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22. Calculating the Confidence Limits
The 95% confidence interval is
We are 95% confident that the mean difference in CPU speed is between and Mhz.
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23. 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
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24. 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
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25. 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
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26. Two Population Proportions
8.3
Confidence Interval Estimation of the Difference Between Two Population Proportions (Large Samples)
Population proportions
Goal: Form a confidence interval for the difference between two population proportions, Px – Py
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27. Two Population Proportions
8.3
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
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28. Two Population Proportions
(continued)
The random variable
is approximately normally distributed
Population proportions
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29. Confidence Interval for Two Population Proportions
The confidence limits for
Px – Py are:
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30. 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
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31. Example: Two Population Proportions
(continued)
Men:
Women:
For 90% confidence, Z/2 = 1.645
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32. 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
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33. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall
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
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