Download Confidence Interval Estimations

Day 3
QMIM
Confidence Interval for a Population mean
of Large and Small Samples
by Binam Ghimire
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Objective
To be able to calculate best estimates of the mean and
standard deviation of a population
To be able to calculate confidence intervals for a
population mean for large and small samples
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Why Sampling? (1)
 Alternate to sampling is to test the entire population.
 The advantage of testing the entire population is the
accuracy
 The disadvantages are 1) Expensive, 2) Time
consuming 3) Destructive and 4) not possible or total
population unknown
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Why Sampling? (2)
 Examples
Car Crash Test
Water Resistant Test – The deep dive watch from
Rolex
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Point Estimates
 Symbols
Population
Sample
Mean
m
Std. deviation
s
s
Size
N
n
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Estimate for the population parameter:
Conditions
Sample should be part of population
Sample should represent the population
Sample should be random
Larger the better
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Confidence Interval Estimations
 Provides Range of Values
 Based on Observations from Sample
 Gives Information about closeness to unknown
population parameter
 Stated in terms of Probability
 90%, 95 %, 99%
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Confidence Interval Estimations
 Probability that the population parameter falls within
certain range
Lower Range
Higher Range
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Confidence Interval Estimations
Lower
Confidence
Limit
Point Estimate
Upper
Confidence
Limit
Width of
confidence interval
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Confidence Interval Estimations
For largest possible sample, margin of error =
0. (This will happen when sample data =
population data)
If so then
m=
But the is from sample so will not be exactly
equal to m. In fact, it will be either lower or
higher than the u [Never 100% (1 – a) ]
m=
+/ - Error ... (1)
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Confidence Interval Estimations:
Central Limit Theorem
We may like to arrive close to m by finding means
from multiple samples. If population is normally
distributed then the “sampling distribution of the
means” will also be normal. If the population is not
normally distributed then whether “sampling
distribution of the mean” will be normal or not
depends on the size of the sample – Central Limit
Theorem.
The spread of the sampling distribution also depends
upon the sample size. Larger the sample size the
narrower the spread (or smaller the standard
deviation)
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Confidence Interval Estimations:
Central Limit Theorem
Source: Oakshott, L. (2006) Essential Quantitative Methods for Business, Management and Finance, 3 rd Ed., Palgrave, p. 226
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Confidence Interval Estimations:
Standard Error
m=
+/ - Error ... (1)
Error depends on two factors: 1) standard deviation
(s) and 2) sample size (n).
We call the error (standard deviation of the sampling
distribution) Standard Error
So we may call the above equation as follows
m=
+/ - Standard Error ... (2)
Standard Error =
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Confidence Interval Estimations:
Standard Error
Standard error of sample mean is the standard
deviation of the distribution of sample means.
When the population σ is known:
When the population σ is unknown:
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Confidence Interval Estimations:
Deriving the Formula
When the number of sample is larger than 30 we can
apply the normal distribution to calculate the limits of
confidence interval (Because Z score table is for
sample size > 30).
Replacing x by , and
get,
by Standard Error ( ) we
... (3)
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Common Levels of confidence:
Checking the Z table
For a 99% confidence Interval, Z = _____
For a 95% confidence Interval, Z = _____
For a 90% confidence Interval, Z = _____
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Exercise
If a sample of 100 items is drawn from a
population and the mean is found to be 200g with
a standard deviation of 5g, find:
(a)a 95% confidence interval estimate for the
population mean.
(b)if a sampling error of only ± 0.5g is allowed,
calculate the size of sample
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Small Samples – T Distribution
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T-Distribution
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Small Samples – T Distribution
 Sample size < 30 = T-distribution
 Properties of t-distribution (Student’s t-distribution)
 Symmetrical (bell shaped)
 Less peaked and fatter tails than a normal
distribution
 Defined by single parameter, degrees of freedom
(df), where df = n – 1
 As df increase, t-distribution approaches normal
distribution
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T table
 The number within the table are t-values not
probabilities
 Numbers in the first column are degree of freedom
(v) of the sample
 This is the freedom that you have in choosing the
values of the sample. If you were given the mean of
the sample of 8 values, you would be free to choose
7 of the 8 but not the 8th one. Therefore there are 7
degrees of freedom
 The number of degrees of freedom is therefore n-1
 For a very large sample the t and Z distributions are
same
 For a 95% confidence interval you will choose 0.025
level
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T-Distribution
 The formula is same like Z, we just replace the Z by t
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Exercise
Sample mean is 494.6 and standard deviation is
23.03. Calculate the confidence interval estimates
for a sample size of 10 at 95% confidence interval
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