day 3 qmim confidence interval for a population mean of large and small samples by binam ghimire
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Day 3 QMIM Confidence Interval for a Population mean of Large and Small Samples by Binam Ghimire. Objective. To be able to calculate best estimates of the mean and standard deviation of a population - PowerPoint PPT PresentationTRANSCRIPT
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Day 3QMIM
Confidence Interval for a Population mean of Large and Small Samples
by Binam Ghimire
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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
Objective
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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)
Car Crash Test
Water Resistant Test – The deep dive watch from Rolex
Examples
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Point Estimates
Population Sample
Mean m
Std. deviation s s
Size N n
Symbols
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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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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%
Confidence Interval Estimations
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Probability that the population parameter falls within certain range
Lower Range Higher Range
Confidence Interval Estimations
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Point EstimateLower Confidence Limit
UpperConfidence Limit
Width of confidence interval
Confidence Interval Estimations
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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 thenm =
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 TheoremWe 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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Source: Oakshott, L. (2006) Essential Quantitative Methods for Business, Management and Finance, 3 rd Ed., Palgrave, p. 226
Confidence Interval Estimations:Central Limit Theorem
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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 =
Confidence Interval Estimations:Standard Error
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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:
Confidence Interval Estimations:Standard Error
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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 by Standard Error ( ) we get,
... (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
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-Distribution
The formula is same like Z, we just replace the Z by t
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ExerciseSample 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