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1
Interval Estimation
UNIVERSITY OF PRETORIA
DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING
ENGINEERING STATISTICS BES221
INTERVAL ESTIMATION
Überhaupt ist es für den Forscher ein guter Morgensport,
täglich vor dem Frühstück eine Lieblingshypothese einzustampfen –
das erhält jung.
Austro-German zoologist
Histograms
Probabilities
Distributions
Distributions
Probability Models
Sampling
Sampling
Estimation
Inference
BASIC CONCEPTS
Point estimates such as the sample estimates :
x for  and s for 
Interval estimates (confidence intervals such as) :
P( x -     x + ) = 1 - 
P S Kruger/2011
Engineering Statistics BES221
2
Interval Estimation
Sampling distributions :
Normal distribution and t-distribution (small sampling theory)
for means, the difference between means, totals and
proportions
The 2- and F-distributions for variances and the difference
between variances
{William Sealy Gossett, Guinness Beer and the Student’s tdistribution}
CONFIDENCE INTERVALS (interval estimates)
General concepts given a sample distribution (assumption)
1-
/2

a
/2

X
b
Sample mean
The probability that the true population mean will be within an
interval (estimate of sample mean plus or minus delta, i.e. between
a and b) may be calculated or if the probability is specified the
confidence limits may be calculated (delta will be a function of the
sample variance and sample size)
P( x -     x + ) = 1 - 
 is known as the level of significance and is usually user specified
P S Kruger/2011
Engineering Statistics BES221
3
Interval Estimation
In similar ways confidence intervals for many other sample
quantities may be calculated, for example the sample variance, the
difference between two sample means, a sample proportion, etc.
Please note : The inherent assumption is usually that the
observations (sample units) are independent. If significant autocorrelation exists between the observations it should be taken into
account, otherwise the confidence limits may be grossly optimistic.
The effect on the “accuracy” of the estimates of :
Increasing or decreasing the sample size
Increasing or decreasing the level of significance
The population variance
CONFIDENCE INTERVALS FOR THE MEAN
For large sample size (n  30)
Maximum error of the estimate of the mean
 = z/2 (  / n )
with z/2 the standardized normal deviate
The sample standard deviation, ( / n), is also often known as the
standard error of the mean
Most often used values for z/2 (from the Standard Normal Table)
z0,005
z0,01
z0,025
z0,05
P S Kruger/2011
= 2,576
= 2,326
= 1,960
= 1,645
Engineering Statistics BES221
4
Interval Estimation
Example :
Sample mean x = 6,25
Sample standard deviation s = 1,60
Sample size n = 40 (“large sample” since n  30)
Calculate a 95% confidence interval
 = 0,05
/2 = 0,025
z/2 = 1,96

= z/2 (  / n )
= 1,96 (1,60 / 40)
= 0,4958
Confidence interval :
P(5,7542    6,7458) = 0,95
Interpretation of the confidence interval :
It should not be interpreted as : “The probability is 0,95 that the
true mean lies within the interval” but rather as : “95% of all
possible intervals calculated according to this procedure will
contain the true mean” !
Calculation of required sample size
n = [ ( z/2 . ) /  ]2
P S Kruger/2011
Engineering Statistics BES221
5
Interval Estimation
Example :
Sample standard deviation s = 1,60
Determine the minimum sample size if the required half-length of
a 95% confidence interval,  = 0,3
Use s as an estimate of 
n
= [ ( z/2 . ) /  ]2
= [ 1,96 . 1,60 / 0,3]2
= 109,2
= 110
For small sample size (n  30)
 = t/2 (  / n )
n = [ ( t/2 . ) /  ]2
Degrees of freedom of the t-distribution :  = n - 1
Extract from a t-table (values of t)

1
5
10
15
20
25
29
30
P S Kruger/2011
0,1
3,078
1,476
1,372
1,341
1,325
1,316
1,311
1,282
0,05
6,314
2,015
1,812
1,753
1,725
1,708
1,699
1,645

0,025
12,706
2,571
2,228
2,131
2,086
2,060
2,045
1,960

0,01
31,821
3,365
2,764
2,602
2,528
2,485
2,462
2,326
0,005
63,657
4,032
3,169
2,947
2,845
2,787
2,756
2,576
1
5
10
15
20
25
29
30
Engineering Statistics BES221
6
Interval Estimation
Confidence Intervals for a Total
Confidence Intervals for a Proportion
Confidence Intervals for a Standard Deviation
Confidence Intervals for the Difference Between Means
Confidence Intervals for the Difference Between Proportions
Confidence Intervals for the Difference Between Means
Sample Size Determination for other Parameters
Independent Samples
Paired Samples
***********************
P S Kruger/2011
Engineering Statistics BES221
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