1. Chapter 7 Estimation: Single Population
Statistics for
Business and Economics 8th Edition
Chapter 7
Estimation: Single Population
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

2. ESTIMATION: AN INTRODUCTION
Definition
The assignment of value(s) to a population parameter based on a value of the corresponding sample statistic is called estimation.
The value(s) assigned to a population parameter based on the value of a sample statistic is called an estimate.
The sample statistic used to estimate a population parameter is called an estimator.
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3. Properties of Unbiased Point Estimators
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

4. POINT AND INTERVAL ESTIMATES
A Point Estimate:
The value of a sample statistic that is used to estimate a population parameter is called a point estimate.
An Interval Estimate:
In interval estimation, an interval is constructed around the point estimate, and it is stated that this interval is likely to contain the corresponding population parameter.
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5. A Point Estimate
Usually, whenever we use point estimation, we calculate the margin of error associated with that point estimation.
The margin of error is calculated as follows:
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6. Interval estimation
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7. Point and Interval Estimates
A point estimate is a single number,
a confidence interval provides additional information about variability
Upper
Confidence
Limit
Lower
Confidence
Limit
Point Estimate
Width of
confidence interval
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

8. Point Estimates μ x P Mean Proportion We can estimate a
Population Parameter …
with a Sample Statistic
(a Point Estimate) μ x
Mean
Proportion P
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

9. Confidence Intervals
Confidence Interval Estimator for a population parameter is a rule for determining (based on sample information) a range or an interval that is likely to include the parameter.
The corresponding estimate is called a confidence interval estimate.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

10. Confidence Interval and Confidence Level
If P(a <  < b) = 1 -  then the interval from a to b is called a 100(1 - )% confidence interval of .
The quantity (1 - ) is called the confidence level of the interval ( between 0 and 1)
In repeated samples of the population, the true value of the parameter  would be contained in 100(1 - )% of intervals calculated this way.
The confidence interval calculated in this manner is written as a <  < b with 100(1 - )% confidence
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

11. Confidence Level, (1-) Suppose confidence level = 95%
(continued)
Suppose confidence level = 95%
Also written (1 - ) = 0.95
A relative frequency interpretation:
From repeated samples, 95% of all the confidence intervals that can be constructed will contain the unknown true parameter
A specific interval either will contain or will not contain the true parameter
No probability involved in a specific interval
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

12. Point Estimate ± (Reliability Factor)(Standard Error)
General Formula
The general formula for all confidence intervals is:
The value of the reliability factor depends on the desired level of confidence
Point Estimate ± (Reliability Factor)(Standard Error)
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

13. Confidence Intervals Confidence Intervals Population Mean Population
Proportion
σ2 Known
σ2 Unknown
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

14. ESTIMATION OF A POPULATION MEAN: Population Variance KNOWN
Three Possible Cases
Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

15. ESTIMATION OF A POPULATION MEAN: Population Variance NOT KNOWN
Three Possible Cases
Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

16. Confidence Interval for μ (σ2 Known)
ESTIMATION OF A POPULATION MEAN: Population Variance KNOWN
Confidence Interval for μ (σ2 Known)
Assumptions
Population variance σ2 is known
Population is normally distributed
If population is not normal, use large sample
Confidence interval estimate:
(where z/2 is the normal distribution value for a probability of /2 in each tail)
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

17. Point Estimate ± (Reliability Factor)(Standard Error)
Margin of Error
Keep in mind that in any time sampling occurs, one expects the possibility of a difference between the particular value of an estimator and the parameter's true value. The true value of an unknown parameter  might be somewhat greater or somewhat less than the value determined by even the best point estimator  .
So, The confidence interval estimate for a parameter takes on the general form-
Can also be written as
where ME is called the margin of error (error factor)
The interval width, w, is equal to twice the margin of error.
w = 2(ME)
The maximum distance between an estimator and the true value of a parameter is called the margin of error.
Point Estimate ± (Reliability Factor)(Standard Error)

18. Reducing the Margin of Error
The margin of error can be reduced if-
The sample size is increased (n↑)
the population standard deviation can be reduced (σ↓)
The confidence level is decreased, (1 – ) ↓
A 95% confidence interval estimate for a population mean is determined to be 62.8 to If the confidence level is reduced to 90%, the confidence interval for becomes narrower
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

19. Finding the Reliability Factor, z/2
Consider a 95% confidence interval:
z = -1.96
z = 1.96
Z units:
Lower
Confidence
Limit
Upper
Confidence
Limit
X units:
Point Estimate
Point Estimate
Find z.025 = 1.96 from the standard normal distribution table
(In 1.96 we find value.9750 from table-1)(So, =.025)( =.95 or,95%)
In the value from table-1 we find 1.96, Hence, z=1.96

20. Common Levels of Confidence
Commonly used confidence levels are 90%, 95%, and 99%
Confidence Coefficient,
Confidence Level
Z/2 value
80%
90%
95%
98%
99%
99.8%
99.9%
.80
.90
.95
.98
.99
.998
.999
1.28
1.645
1.96
2.33
2.58
3.08
3.27
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

21. Intervals and Level of Confidence
Sampling Distribution of the Mean x
Intervals extend from to x1
100(1-)% of intervals constructed contain μ;
100()% do not. x2
Confidence Intervals
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

22. Figure 8.4 Confidence intervals.
Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

23. Example-1: Refined Sugar (Confidence Interval)
A process produces bags of refined sugar. The weights of the content of these bags are normally distributed with standard deviation 1.2 ounces. The contents of a random sample of 25 bags has a mean weight of 19.8 ounces.
Find the upper and lower confidence limits of a 99% confidence interval for the true mean weight for all bags of sugar produced by the process.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

24. Example 2 (practice)
A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.
Determine a 90% confidence interval for the true mean resistance of the population.
Here, Confidence level is 90% or .90, so The area in each tail of the normal distribution curve is α/2=(1-.90)/2=.05
In the value from table-1 we find 1.65 Hence, z = 1.65
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

25. Example-2 (practice) Solution:
(continued)
A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is .35 ohms.
Solution:
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

26. Example 2(practice)Interpretation
We are 90% confident that the true mean resistance is between and ohms
Although the true mean may or may not be in this interval, 90% of intervals formed in this manner will contain the true mean
Also See Ex-7.3 from book
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

27. Figure 8.4 Confidence intervals.
Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

28. ESTIMATION OF A POPULATION MEAN: Population Variance NOT KNOWN
Three Possible Cases
Prem Mann, Introductory Statistics, 7/E Copyright © 2010 John Wiley & Sons. All right reserved

29. Confidence Intervals Confidence Intervals Population Mean Population
Proportion
σ2 Known
σ2 Unknown
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

30. Student’s t Distribution
Consider a random sample of n observations
with mean x and standard deviation s
from a normally distributed population with mean μ
Then the variable
follows the Student’s t distribution with (n - 1) degrees of freedom
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

31. Confidence Interval for μ (σ2 Unknown)
If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s
This introduces extra uncertainty, since s is variable from sample to sample
So we use the t distribution instead of the normal distribution
The Student’s t distribution is the ratio of the standard normal distribution to the square root of the chi-square distribution divided by its degrees of freedom.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

32. Confidence Interval for μ (σ Unknown)
(continued)
Assumptions
Population standard deviation is unknown
Population is normally distributed
If population is not normal, use large sample
Use Student’s t Distribution
Confidence Interval Estimate:
where tn-1,α/2 is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail:
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

33. Student’s t Distribution
The t is a family of distributions
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
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

34. Student’s t Table-8 (appendix)
Upper Tail Area
Let: n = df = n - 1 =  = /2 =.05 df
.10
.05
.025 1
3.078
6.314
12.706 2
1.886
2.920
4.303
/2 = .05 3
1.638
2.353
3.182
The body of the table contains t values, not probabilities t
2.920
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

35. Example 7.4 (book)
Gasoline prices rose drastically during the early years of this century. Suppose that a recent study was conducted using truck drivers with equivalent years of experience to test run 24 trucks of a particular model over the same highway. The sample mean and standard deviation is and respectively.
Estimate the population mean fuel consumption for this truck model with 90% confidence interval.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

36. Example-3 (practice) s = 8. Form a 95% confidence interval for μ
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
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

37. Confidence Intervals Confidence Intervals Population Mean Population
Proportion
σ Known
σ Unknown
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

38. Confidence Intervals for the Population Proportion, p
An interval estimate for the population proportion ( P ) can be calculated by adding an allowance for uncertainty to the sample proportion ( )
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

39. Confidence Intervals for the Population Proportion, p
(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:
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

40. Confidence Interval Endpoints
Upper and lower confidence limits for the population proportion are calculated with the formula
where
z/2 is the standard normal value for the level of confidence desired
is the sample proportion
n is the sample size
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

41. Example-4 A random sample of 100 people shows that 25 are left-handed.
Form a 95% confidence interval for the true proportion of left-handers
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

42. Example-4
(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.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.

43. Ex-4 Interpretation
We are 95% confident that the true percentage of left-handers in the population is between
16.51% and 33.49%.
Although the interval from to 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.
Statistics for Business and Economics, 7e © 2007 Pearson Education, Inc.