Estimation of Allele Frequencies from Quantitative Trait Data
QM-1/2011/Estimation Page 1 Quantitative Methods Estimation.
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Transcript of QM-1/2011/Estimation Page 1 Quantitative Methods Estimation.
QM-1/2011/EstimationPage 1
Quantitative MethodsEstimation
QM-1/2011/EstimationPage 2
Estimation Process
PopulationPopulation
Mean, , is unknown
Random SampleRandom Sample
Mean X= 50 Are we
confident that is
50?
OR we are more confident in saying that is between 48
& 52?
QM-1/2011/EstimationPage 3
Population Parameters are Estimated
Population Parameter Sample Statistic
2
xi
n
(xi-x)2
n-1
QM-1/2011/EstimationPage 4
Estimation Methods
Estimation
Point Estimation Interval Estimation
QM-1/2011/EstimationPage 5
Estimation Methods
Estimation
Point Estimation Interval Estimation
QM-1/2011/EstimationPage 6
Point Estimation Point Estimate
Draw a sample from a population and compute AVERAGE to ESTIMATE the MEAN Value of the population.
The formula used for computing Average is called ‘Statistic’ or ‘Estimator’.
The computed value based on the sample is called ‘Estimate’ for the population parameter (here ‘Mean’).
QM-1/2011/EstimationPage 7
Problem with Point Estimate Following data is a sample from a
population. 13,13,16,19,17,15,14,13,15,15,18. Point estimate for Mean is simply the
average of these data, i.e. 15.27. If we had collected lesser or larger
number of samples the average would have been different.
QM-1/2011/EstimationPage 8
Problem with Point Estimate
Population Parameter
1st groupof sample
2nd groupof sample
Estimates from different samples will be different,
but, within a limit, around the Population
parameter. This is applicable for all estimates (i.e.
Average, s.d. etc). All estimates from different
samples are ‘Point estimates’.
QM-1/2011/EstimationPage 9
Estimation Methods
Estimation
Point Estimation Interval Estimation
QM-1/2011/EstimationPage 10
Interval Estimate This estimate gives a lower and upper
limit. The population parameter will be in this interval with a certain degree of confidence.A confidence level is associated with an
interval estimate.Higher the confidence level desired, the
spread of the interval will be larger.
QM-1/2011/EstimationPage 11
Error in Estimate
Sample statistic Sample statistic
(point estimate)(point estimate)
Population Parameter Population Parameter (unknown)(unknown)
x
X ~ N(, /√n)
~ N(0, 1)X – /√n
QM-1/2011/EstimationPage 12
Standard Normal & Tail area
0.95 0.05
Z 0.05
-4 -3 -2 -1 0 1 2 3 4
P(Z < Z0.05) = 0.95
QM-1/2011/EstimationPage 13
Standard Normal & Tail area
(1- )
Z
-4 -3 -2 -1 0 1 2 3 4
QM-1/2011/EstimationPage 14
Standard Normal & 2-Tail area
-4 -3 -2 -1 0 1 2 3 4
(1- )
Z
QM-1/2011/EstimationPage 15
Estimation of Interval P(|Z |< Z0.025) = 0.95
P(|X – | < Z0.025) = 0.95
P(|X – | < Z0.025 /√n) = 0.95
P(-Z0.025 /√n < - X < Z0.025 /√n) = 0.95
P(X - Z0.025 /√n < < X + Z0.025 /√n) = 0.95
/√n
QM-1/2011/EstimationPage 16
Confidence Interval for Mean P(X – Z/2 /√n < < X + Z/2 /√n) = 1-
(1-) /2 Z/2
0.80 0.20 0.10 1.28
0.90 0.10 0.05 1.65
0.95 0.05 0.025 1.96
0.98 0.02 0.01 2.33
0.99 0.01 0.005 2.58
QM-1/2011/EstimationPage 17
Wider interval for higher confidence
99%99%
xx - 2.58- 2.58xxxx + 2.58+ 2.58xx
95%95%
x -1.96x -1.96xx x +1.96x +1.96xx
90%90%
x -1.65x -1.65xx x +1.65x +1.65xx
QM-1/2011/EstimationPage 18
Factors Affecting Interval Width Data Dispersion
Measured by Higher results wider interval.
Sample SizeX = n
Higher sample size results narrower interval.
Level of Confidence (1 - )Affects ZHigher level of confidence, wider interval.
QM-1/2011/EstimationPage 19
Confidence Interval Estimates
Confidence Interval
Mean Proportion
Known Unknown
QM-1/2011/EstimationPage 20
Confidence Interval Estimates
Confidence Interval
Mean Proportion
Known Unknown
QM-1/2011/EstimationPage 21
Confidence Interval of Mean - known
Population Standard Deviation is Known Population is Normally Distributed. n > 30, if Population is Not Normal. Confidence Interval:
nZX
nZX
2/2/
QM-1/2011/EstimationPage 22
Exercise The mean of a random sample of n = 36
isX = 50. Set up a 95% confidence interval estimate for if = 12.
QM-1/2011/EstimationPage 23
Exercise You’re a Q/C inspector of Coke. The for
2-liter bottles is 0.05 liters. A random sample of 100 bottles showedX = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles?
QM-1/2011/EstimationPage 24
Confidence Interval Estimates
Confidence Interval
Mean Proportion
Known Unknown
QM-1/2011/EstimationPage 25
Confidence Interval of Mean - Unknown
Normal population. n > 30 The sample sd s is a good estimator of Confidence interval is as below.
nZX
nZX
ss 2/2/