1. Chapter 8 Estimating Single Population Parameters
Business Statistics:
A Decision-Making Approach
8th Edition
Chapter 8 Estimating Single Population Parameters

2. Chapter Goals After completing this chapter, you should be able to:
Distinguish between a point estimate and a confidence interval estimate
Construct and interpret a confidence interval estimate for a single population mean using both the z and t distributions
Determine the required sample size to estimate a single population mean or a proportion within a specified margin of error
Form and interpret a confidence interval estimate for a single population proportion

3. Overview of the Chapter
Builds upon the material from Chapter 1 and 7
Introduces using sample statistics to estimate population parameters
Because gaining access to population parameters can be expensive, time consuming and sometimes not feasible
Confidence Intervals for the Population Mean, μ
when Population Standard Deviation σ is Known
when Population Standard Deviation σ is Unknown
Confidence Intervals for the Population Proportion, p
Determining the Required Sample Size for means and proportions

4. Estimation Process Population
Confidence Level
Random Sample Mean
(point estimate)
I am 95% confident that μ is between 40 & 60.
Population
Mean
x = 50
(mean, μ, is unknown)
Sample
confidence
interval

5. Point Estimate
Suppose a poll indicate that 62% (sample mean) of the people favor limiting property taxes to 1% of the market value of the property.
The 62% is the point estimate of the true population of people who favor the property-tax limitation.
EPA tested average Automobile Mileage (point estimate)

6. Confidence Interval
The point estimate (sample mean) is not likely to exactly equal the population parameter because of sampling error.
Probability of “sample mean = population mean” is zero
Problem: how far the sample mean is from the population mean.
To overcome this problem, “confidence interval” can be used as the most common procedure.
An interval developed from sample values such that if all possible intervals of a given width were constructed, a percentage of these intervals, known as the confidence level, would include the true population parameter.

7. Point and Interval Estimates
So, a confidence interval provides additional information about variability within a range.
The interval incorporates the sampling error
Lower
Confidence
Limit
Upper
Confidence
Limit
Point Estimate
Width of
confidence interval

8. Confidence Level Confidence level = 95%
Meaning: In the long run, 95% of all the confidence intervals will contain the true parameter
Describes how strongly we believe that a particular sampling method will produce a confidence interval that includes the true population parameter.
A percentage (less than 100%)
Most common: 90% (α = 0.1), 95% (α = 0.05), 99%

9. Common Levels of Confidence
Commonly used confidence levels are 90%, 95%, and 99%
Confidence Level
Critical value, z
80%
90%
95%
98%
99%
99.8%
99.9%
1.28
1.645
1.96
2.33
2.58
3.08
3.27

10. Point Estimate  Margin of Error (z or t - Value) * (Standard Error)
General Formula
The general formula for the confidence interval is:
Point Estimate  Margin of Error
Margin of Error: Margin of Error (e): the amount added and subtracted to the point estimate to form the confidence interval
(z or t - Value) * (Standard Error)

11. Point Estimate  (Critical Value)(Standard Error)
General Formula
According to our textbook
Point Estimate  (Critical Value)(Standard Error)
z or t -value based on the level of confidence desired
Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

12. Conditions for finding the Critical Value
The Central Limit Theorem states that the sampling distribution of a statistic will be normally (nearly normally) distributed if at least n > 30.
The sampling distribution of the mean is normally distributed when the population distribution is normally distributed.

13. How to Find the z Value
When one of these conditions is satisfied, the critical value can be expressed as a z score.
To find the z value, see the next slide.

14. Finding the z Value -z = -1.96 z = 1.96
Consider a 95% confidence interval:
0.95/2 (because of lower and upper limit) = find on z table = 1.96
-z = -1.96
z = 1.96
z units:
Lower
Confidence
Limit
Upper
Confidence
Limit
x units:
Point Estimate
Point Estimate

15. How to Find the t value d.f. = n – 1
When the population distribution type is unknown or when the sample size is small (less than 30), the t value is preferred.
The t value depends on degrees of freedom (d.f.)
Number of observations that are free to vary after sample mean has been calculated
d.f. = n – 1
Only n-1 independent pieces of data information left in the sample because the sample mean has already been obtained

16. t Distribution: Try the t simulation one the website
Note: t compared to z as n increases
As n the estimate of s becomes better so t converges to z
Standard Normal
(t with df = )
t (df = 13)
t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
t (df = 5) t

17. t Table 0.90 2 t /2 = 0.05 2.920 Confidence Level df 0.50 0.80 1
Let: n = df = n - 1 = 2 confidence level: 90%
0.90 df
0.50
0.80 1
1.000
3.078
6.314 2
0.817
1.886
2.920
/2 = 0.05 3
0.765
1.638
2.353
The body of the table contains t values, not probabilities t
2.920

18. With comparison to the z value
t Distribution Values
With comparison to the z value
Note: t compared to z as n increases
Confidence t t t z
Level (10 d.f.) (20 d.f.) (30 d.f.) ____

19. Example A random sample of n = 25 has x = 50 and
s = 8. Form a 95% confidence interval for μ
d.f. = n – 1 = 24, so
The confidence interval is

20. Summary of When to use z or t
Use z value (or score)
Population is normally distributed (very rare and even though sample size is small)
Sample size is n > 30
Use t value (or score)
Do not know population distribution type (not normal)
Do not know Pop. Mean and Std
Many real world situations
Sample size is less than 30

21. Using Analysis ToolPak
Download “Confidence Interval Example” Excel file

22. Using Analysis ToolPak
small sample: apply “t” distribution automatically)
Confidence Interval
119.9 (mean) ± 2.59
Don’t even worry about p* = 1 - α/2
Margin of Error

23. Using Analysis ToolPak (large sample: use normal (z) distribution automatically )

24. Determining Required Sample Size
Wishful situation
High confidence level, low margin of error, and right sample size
In reality, conflict among three….
For a given sample size, a high confidence level will tend to generate a large margin of error
For a given confidence level, a small sample size will result in an increased margin of error
Reducing of margin of error requires either reducing the confidence level or increasing the sample size, or both

25. Determining Required Sample Size
How large a sample size do I really need?
Sampling budget constraint
Cost of selecting each item in the sample

26. Determining Required Sample Size When σ is known
The required sample size can be found to reach a desired margin of error (e) and
level of confidence (1 - )
Required sample size, σ known:

27. Required Sample Size Example
If  = 45 (known), what sample size is needed to be 90% confident of being correct within ± 5?
So the required sample size is n = 220
(Always round up)

28. If σ is unknown (most real situation)
If σ is unknown, three possible approaches
Use a value for σ that is expected to be at least as large as the true σ
Select a pilot sample (smaller than anticipated sample size) and then estimate σ with the pilot sample standard deviation, s
Use the range of the population to estimate the population’s Std Dev. As we know, µ ± 3σ contains virtually all of the data. Range = max – min.
Thus, R = (µ + 3σ) – (µ - 3σ) = 6σ. Therefore, σ = R/6 (or R/4 for a more conservative estimate, producing a larger sample size)

29. Example when σ is unknown
Example 8-5: Jackson’s Convenience Stores
Using a pilot sample approach
Available on the class website
Make sure to review all the examples from page 306 to 327.

30. Confidence Intervals for the Population Proportion, π
Try by yourself!
An interval estimate for the population proportion ( π ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p ).
For example, estimation of the proportion of customers who are satisfied with the service provided by your company
Sample proportion, p = x/n

31. Confidence Intervals for the Population Proportion, π
(continued)
Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation
We will estimate this with sample data:
See Chpt. 7!!

32. Confidence Interval Endpoints
Upper and lower confidence limits for the population proportion are calculated with the formula
where
z is the standard normal value for the level of confidence desired
p is the sample proportion
n is the sample size

33. Example A random sample of 100 people shows that 25 are left-handed.
Form a 95% confidence interval for the true proportion of left-handers

34. Example
(continued)
A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers. 1. 2. 3.

35. Interpretation
We are 95% confident that the true percentage of left-handers in the population is between
16.51% and 33.49%
Although this range may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.

36. Changing the sample size
Increases in the sample size reduce the width of the confidence interval.
Example:
If the sample size in the above example is doubled to 200, and if 50 are left-handed in the sample, then the interval is still centered at 0.25, but the width shrinks to

37. Finding the Required Sample Size for Proportion Problems
Define the
margin of error:
Solve for n:
Will be in % units
π can be estimated with a pilot sample, if necessary (or conservatively use π = 0.50 – worst possible variation thus the largest sample size)

38. What sample size...?
How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence?
(Assume a pilot sample yields p = 0.12)

39. What sample size...? Solution: For 95% confidence, use Z = 1.96
(continued)
Solution:
For 95% confidence, use Z = 1.96
e = 0.03
p = 0.12, so use this to estimate π
So use n = 451

40. Using PHStat
PHStat can be used for confidence intervals for the mean or proportion
Two options for the mean: known and unknown population standard deviation
Required sample size can also be found
Download from the textbook website
The link available on the class website

41. PHStat Interval Options

42. PHStat Sample Size Options

43. Using PHStat (for μ, σ unknown)
A random sample of n = 25 has x = 50 and
s = 8. Form a 95% confidence interval for μ

44. Using PHStat (sample size for proportion)
How large a sample would be necessary to estimate the true proportion defective in a large population within 3%, with 95% confidence?
(Assume a pilot sample yields p = 0.12)

45. Chapter Summary Discussed point estimates
Introduced interval estimates
Discussed confidence interval estimation for the mean [σ known]
Discussed confidence interval estimation for the mean [σ unknown]
Addressed determining sample size for mean and proportion problems
Discussed confidence interval estimation for the proportion

46. Printed in the United States of America.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.
Printed in the United States of America.