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Слайды и текст к этой презентации:
№1 слайд
Содержание слайда: Confidence Interval Estimation
№2 слайд
Содержание слайда: Types of Estimates
Point Estimate
A single number used to estimate an unknown population parameter
Interval Estimate
A range of values used to estimate a population parameter
Characteristics
Better idea of reliability of estimate
Decision making is facilitated
№3 слайд
Содержание слайда: Point Estimates
№4 слайд
Содержание слайда: Point and Interval Estimates
A point estimate is a single number,
a confidence interval provides additional information about variability
№5 слайд
Содержание слайда: Confidence Interval Estimate
An interval gives a range of values:
Takes into consideration the variation in sample statistics from sample to sample
Based on observation from 1 sample
Gives information about closeness to unknown population parameters
Stated in terms of level of confidence
Can never be 100% confident
№6 слайд
Содержание слайда: Confidence Level, (1-)
Suppose confidence level γ = 95%
Also written γ =(1 - ) = .95
Where is the risk of being wrong
A relative frequency interpretation:
In the long run, 95% of all the confidence intervals that can be constructed will contain the unknown parameter
A specific interval either will contain or will not contain the true parameter
No probability involved in a specific interval
№7 слайд
Содержание слайда: Estimation Process
№8 слайд
Содержание слайда: General Formula
The general formula for all confidence intervals is:
№9 слайд
Содержание слайда: Confidence Intervals
№10 слайд
Содержание слайда: Confidence Interval for μ
(σ Known)
Assumptions
Population standard deviation σ is known
Population is normally distributed
If population is not normal, use large sample
Confidence interval estimate for μ
№11 слайд
Содержание слайда: Finding the Critical Value
№12 слайд
Содержание слайда: Finding the Critical Value
Consider a 95% confidence interval:
№13 слайд
Содержание слайда: Margin of Error
Margin of Error (e): the amount added and subtracted to the point estimate to form the confidence interval
№14 слайд
Содержание слайда: Factors Affecting
Margin of Error
Data variation, σ : e as σ
Sample size, n : e as n
Level of confidence, 1 - : e if γ =1 -
№15 слайд
Содержание слайда: Example
Example
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.
Determine a 95% confidence interval for the true mean resistance of the population.
№16 слайд
Содержание слайда: Solution –
№17 слайд
Содержание слайда: Interpretation
We are γ=95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms
№18 слайд
Содержание слайда: Confidence Interval for μ
(σ Unknown)
If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s as an estimate
In these case the t-distribution is used instead of the normal distribution
№19 слайд
Содержание слайда: Student’s t Distribution
№20 слайд
Содержание слайда: Confidence Interval for μ
(σ Unknown)
Assumptions
Population standard deviation is unknown
Population is not highly skewed
Population is normally distributed or the sample size is large (>30)
Use Student’s t Distribution
№21 слайд
Содержание слайда: Confidence Interval Estimate:
where t is the critical value of the t-distribution with n-1 degrees of freedom and an area of α/2 in each tail)
№22 слайд
Содержание слайда: Define t from equation
– Confidence Coefficient.
t - obtain with using Excel function TINV.
t = TINV(1- γ; n-1)
=T.INV.2T (1- γ; n-1)
№23 слайд
Содержание слайда: Example
A random sample of n = 25 has X = 50 and
S = 8. Form 95% confidence interval for μ
degrees of freedom = n – 1 = 24,
=0,95.
№24 слайд
Содержание слайда: To get a t - value use the TINV function.
To get a t - value use the TINV function.
The value of alpha =(1-confidence) and
n-1 degrees of freedom are the inputs needed.
For 95% confidence use alpha =0.05 and for a sample size of 25 use 24 df
t= TINV(0,05; 24)=2,0639
t= T.INV.2T(0,05;24)= 2,0639
№25 слайд
№26 слайд
Содержание слайда: Confidence Interval on the Variance and Standard Deviation of a Normal Distribution
№27 слайд
Содержание слайда: Confidence Interval on the Variance and Standard Deviation of a Normal Distribution
№28 слайд
№29 слайд
Содержание слайда: Confidence Interval on the Variance and Standard Deviation of a Normal Distribution
№30 слайд
№31 слайд
Содержание слайда: We can use 2 –Table for solving next equation
Or EXCEL function CHIINV (q; n-1),
=CHISQ.INV.RT(q;n-1).
№32 слайд
Содержание слайда: EXAMPLE
According to the 20 measurements found standard deviation S = 0,12. Find precision measurements with reliability 0.98.
№33 слайд
Содержание слайда: With using CHIINV (q; n-1) we obtain 12 і 22 .
With using CHIINV (q; n-1) we obtain 12 і 22 .
For degrees of freedom n - 1=19 and probability α2=(1-0,98)/2=0,01 define
22 =36,2,
after that for n - 1=19 and probability α1=(1+0,98)/2=0,99 define 12 =7,63.
22 = CHIINV(0,01; 19)=36,2 ; =CHISQ.INV.RT(0,01;19).
12 = CHIINV(0,99;19)=7,63.
=CHISQ.INV.RT(0,01;19).
№34 слайд
Содержание слайда: calculate
Confidence Interval for σ is (0,09;0,19).