1. Statistics for Business and Economics
Estimation of Population Parameters: Confidence Intervals Chapter 8

2. Learning Objectives 1. State What Is Estimated
2. Distinguish Point & Interval Estimates
3. Explain Interval Estimates
4. Compute Confidence Interval Estimates for Population Mean & Proportion
5. Compute Sample Size
As a result of this class, you will be able to ...

3. Thinking Challenge
Suppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out?

4. Statistical Methods Statistical Methods Descriptive Inferential
Statistics
Statistics
Hypothesis
Estimation
Testing

5. I am 95% confident that mX is between 40 & 60.
Estimation Process
Population
Random Sample
I am 95% confident that mX is between 40 & 60.
Mean `X = 50
Mean, mX, is unknown

6. Unknown Population Parameters Are Estimated
Estimate Population
with Sample
Parameter...
Statistic
Mean m ` x x
Proportion p p s 2
Variance 2 s s x
Differences m
- m ` x - ` x 1 2 1 2

7. Estimation Methods Estimation Point Interval Estimation Estimation
Confidence
Boot-
Prediction
Interval
strapping
Interval

8. Point Estimation 1. Provides Single Value
Based on Observations from 1 Sample
2. Gives No Information about How Close Value Is to the Unknown Population Parameter
3. Sample Mean`X = 3 Is Point Estimate of Unknown Population Mean

9. Estimation Methods Estimation Point Interval Estimation Estimation
Confidence
Boot-
Prediction
Interval
strapping
Interval

10. Interval Estimation 1. Provides Range of Values
Based on Observations from 1 Sample
2. Gives Information about Closeness to Unknown Population Parameter
Stated in terms of Probability
Knowing Exact Closeness Requires Knowing Unknown Population Parameter
3. e.g., Unknown Population Mean Lies Between 50 & 70 with 95% Confidence

11. Key Elements of Interval Estimation
A Probability That the Population Parameter Falls Somewhere Within the Interval.
Sample Statistic (Point Estimate)
Confidence Interval
Confidence Limit (Lower)
Confidence Limit (Upper)

12. Confidence Limits for Population Mean
Parameter = Statistic ± Error
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13. Many Samples Have Same Confidence Interval
`X = mx ± Zs`x s x _ `X
mx-2.58s`x mx-1.65s`x mx
mx+1.65s`x mx+2.58s`x
mx-1.96s`x
mx+1.96s`x
90% Samples
95% Samples
99% Samples

14. Level of Confidence
1. Probability that the Unknown Population Parameter Falls Within Interval
2. Denoted (1 - a) %
a Is Probability That Parameter Is Not Within Interval
3. Typical Values Are 99%, 95%, 90%

15. Intervals & Level of Confidence_
Sampling Distribution of Mean s x a /2 a /2
1 - a _
Notice that the interval width is determined by 1-a in the sampling distribution. X m = m ` x x
Intervals Extend from `X - Zs`X to `X + Zs`X
(1 - a) % of Intervals Contain mX .
a % Do Not.
Large Number of Intervals

16. Factors Affecting Interval Width
1. Data Dispersion
Measured by sX
2. Sample Size
s`X = sX / Ön
3. Level of Confidence (1 - a)
Affects Z
Intervals Extend from `X - Zs`X to`X + Zs`X
Have students explain why each of these occurs.
Level of confidence can be seen in the sampling distribution.
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17. Confidence Interval Estimates
Intervals
Mean
Proportion
Variance
Finite s
Known s
Unknown x x
Population

18. Confidence Interval Estimate Mean (sX Known)
1. Assumptions
Population Standard Deviation Is Known
Population Is Normally Distributed
If Not Normal, Can Be Approximated by Normal Distribution (n ³ 30)
2. Confidence Interval Estimate s s X X X - Z × m X + Z × a / 2 X a / 2 n n

19. Estimation Example Mean (sX Known)
The mean of a random sample of n = 25 is`X = 50. Set up a 95% confidence interval estimate for mX if sX = 10. s s X X X - Z × m X + Z × a / 2 X a / 2 n n 10 10 50 - 1 . 96 × m 50 + 1 . 96 × X 25 25 46 . 08 m 53 . 92 X

20. Thinking Challenge
You’re a Q/C inspector for Gallo. The sX for 2-liter bottles is .05 liters. A random sample of 100 bottles showed`X = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles?
2 liter
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21. Confidence Interval Solution*X X X - Z × m X + Z × a / 2 X a / 2 n n . 05 . 05 1 . 99 - 1 .
645 × m 1 . 99 + 1 .
645 × X
100
100 1 .
982 m 1 .
998 X

22. Confidence Interval Estimates
Intervals
Mean
Proportion
Variance
Finite s
Known s
Unknown x x
Population

23. Confidence Interval Estimate Mean (sX Unknown)
1. Assumptions
Population Standard Deviation Is Unknown
Population Must Be Normally Distributed
2. Use Student’s t Distribution
3. Confidence Interval Estimate S S X - t × m X + t × a / 2 , n - 1 X a / 2 , n - 1 n n

24. Student’s t Distribution
Standard Normal
Bell-Shaped
Symmetric
‘Fatter’ Tails
t (df = 13)
t (df = 5) Z t

25. Student’s t Table .05 2 t a / 2 .05 2.920 t Values Upper Tail Area df
Assume: n = 3 df = n - 1 = 2 a = .10 a/2 =.05
a / 2
Upper Tail Area df
.25
.10
.05
Confidence intervals use a/2, so divide a! 1
1.000
3.078
6.314 2
0.817
1.886
2.920
.05 3
0.765
1.638
2.353 t
2.920
t Values

26. Degrees of Freedom (df)
1. Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated
2. Example
Sum of 3 Numbers Is 6 X = 1 (or Any Number) X = 2 (or Any Number) X = 3 (Cannot Vary) Sum = 6
degrees of freedom = n -1 = 3 -1 = 2

27. Estimation Example Mean (sX Unknown)
A random sample of n = 25 has`X = 50 & S = 8. Set up a 95% confidence interval estimate for mX. S S X - t × m X + t × a / 2 , n - 1 X a / 2 , n - 1 n n 8 8 50 - 2 .
0639 × m 50 + 2 .
0639 × X 25 25 46 . 69 m 53 . 30 X

28. Thinking Challenge
You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1.
What is the 90% confidence interval estimate of the population mean task time?
Allow students about 20 minutes to solve.

29. Confidence Interval Solution*
`X = 3.7
S =
n = 6, df = n -1 = = 5
S / Ön = / Ö6 =
t.05,5 =
3.7 - (2.015)(1.592) £ mX £ (2.015)(1.592)
.492 £ mX £ 6.908

30. Confidence Interval Estimates
Intervals
Mean
Proportion
Variance
Finite s
Known s
Unknown x x
Population

31. Estimation for Finite Populations
1. Assumptions
Sample Is Large Relative to Population
n / N > .05
2. Use Finite Population Correction Factor
3. Confidence Interval (Mean, sX Unknown) S N - n S N - n X - t × × m X + t × × a / 2 , n - 1 X a / 2 , n - 1 n N - 1 n N - 1

32. Confidence Interval Estimates
Intervals
Mean
Proportion
Variance
Finite s
Known s
Unknown x x
Population

33. Confidence Interval Estimate Proportion
1. Assumptions
Two Categorical Outcomes
Population Follows Binomial Distribution
Normal Approximation Can Be Used
n·p ³ 5 & n·(1 - p) ³ 5
2. Confidence Interval Estimate p × ( 1 - p ) p × ( 1 - p ) s s s s p - Z × p p + Z × s s n n

34. Estimation Example Proportion
A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p. p × ( 1 - p ) p × ( 1 - p ) s s s s p - Z × p p + Z × s a / 2 s a / 2 n n . 08 × ( 1 - . 08 ) . 08 × ( 1 - . 08 ) . 08 - 1 . 96 × p . 08 + 1 . 96 ×
400
400 .
053 p .
107

35. Thinking Challenge
You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?

36. Confidence Interval Solution*ü
n·p ³ 5
n·(1 - p) ³ 5 p × ( 1 - p ) p × ( 1 - p ) s s s s p - Z × p p + Z × s a / 2 s a / 2 n n .
175 × (.
825 ) .
175 × (.
825 ) .
175 - 1 .
645 × p .
175 + 1 .
645 ×
200
200 .
1308 p .
2192

37. Estimation Methods Estimation Point Interval Estimation Estimation
Confidence
Boot-
Prediction
Interval
strapping
Interval

38. Bootstrapping Method 1. Used If Population Is Not Normal
2. Requires Computer
3. Steps
Take Initial Sample
Sample Repeatedly from Initial Sample
Compute Sample Statistic
Form Resampling Distribution
Limits Are Values That Cut Off Smallest & Largest a/2 %

39. Estimation Methods Estimation Point Interval Estimation Estimation
Confidence
Boot-
Prediction
Interval
strapping
Interval

40. Prediction Interval 1. Used to Estimate Future Individual X Value
2. Not Used to Estimate Unknown Population Parameter
3. Prediction Interval Estimate

41. I don’t want to sample too much or too little!
Finding Sample Sizes
I don’t want to sample too much or too little! X - m
Error x
(1) Z = = s s x x s x
(2)
Error = Z s = Z x n 2 2 Z s x ( 3 ) n = 2
Error

42. ( ) ( ) ( ) Sample Size Example
What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard deviation is 45. ( ) ( ) 2 2 2 2 Z s 1 .
645 45 x n = = ( ) =
219 . 2 @
220 2 2
Error 5

43. Thinking Challenge
You work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that sX was about $400. What sample size do you use?

44. ( ) ( ) ( ) Sample Size Solution* Z s n = Error 1 . 96 400 = 50 = 245x n = 2
Error ( ) ( ) 2 2 1 . 96
400 = ( ) 2 50 =
245 . 86 @
246

45. Conclusion 1. Stated What Is Estimated
2. Distinguished Point & Interval Estimates
3. Explained Interval Estimates
4. Computed Confidence Interval Estimates for Population Mean & Proportion
5. Computed Sample Size

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End of Chapter
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