1. Chapter 8 Estimation: Additional Topics
Statistics for
Business and Economics 7th Edition
Chapter 8
Estimation: Additional Topics
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2. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
Determine the required sample size to estimate a mean or proportion within a specified margin of error
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3. Estimation: Additional Topics
Chapter Topics
Confidence Intervals
Population Means, Dependent Samples
Population Means, Independent Samples
Sample Size Determination
Population Proportions
Examples:
Large Populations
Finite Populations
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
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
Dependent samples
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 difference between 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
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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11. 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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12. * 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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13. 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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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
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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
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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
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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
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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
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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 74 56
Assume both populations are
normal with equal variances, and use 95% confidence
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20. Calculating the Pooled Variance
The pooled variance is:
The t value for a 95% confidence interval is:
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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.
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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
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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
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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
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25. Two Population Proportions
8.3
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
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26. Two Population Proportions
(continued)
The random variable
is approximately normally distributed
Population proportions
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27. Confidence Interval for Two Population Proportions
The confidence limits for
Px – Py are:
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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
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29. Example: Two Population Proportions
(continued)
Men:
Women:
For 90% confidence, Z/2 = 1.645
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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
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31. Sample Size Determination
8.4
Determining
Sample Size
Large Populations
Finite Populations
For the
Mean
For the
Proportion
For the
Mean
For the
Proportion
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32. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
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
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33. Sample Size Determination
Large Populations
For the
Mean
Margin of Error (sampling error)
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34. Sample Size Determination
(continued)
Large Populations
For the
Mean
Now solve for n to get
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35. 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 population standard deviation, σ
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36. 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)
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37. Sample Size Determination: Population Proportion
Large Populations
For the
Proportion
Margin of Error (sampling error)
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38. Sample Size Determination: Population Proportion
(continued)
Large Populations
For the
Proportion
cannot be larger than 0.25, when = 0.5
Substitute 0.25 for
and solve for n to get
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39. Sample Size Determination: Population Proportion
(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
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40. Required Sample Size Example: Population Proportion
How large a sample would be necessary to estimate the true proportion defective in a large population within ±3%, with 95% confidence?
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41. 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
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42. Sample Size Determination: Finite Populations
8.5
Finite Populations
Calculate the required sample size n0 using the prior formula:
Then adjust for the finite population:
For the
Mean
A finite population correction factor is added:
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43. Sample Size Determination: Finite Populations
Solve for n:
The largest possible value for this expression
(if P = 0.25) is:
3. A 95% confidence interval will extend ± from the sample proportion
For the
Proportion
A finite population correction factor is added:
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44. Example: Sample Size to Estimate Population Proportion
(continued)
How large a sample would be necessary to estimate within ±5% the true proportion of college graduates in a population of 850 people with 95% confidence?
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45. Required Sample Size Example
(continued)
Solution:
For 95% confidence, use z0.025 = 1.96
ME = 0.05
So use n = 265
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46. Copyright © 2010 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
Formed confidence intervals for the population variance using the chi-square distribution
Determined required sample size to meet confidence and margin of error requirements
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