10주차
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Introduction to Probability and Statistics10th Week (5/17)
Estimation Theory
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Confidence Intervals
Content of this chapter Confidence Intervals for the Population
Mean, μ when Population Standard Deviation σ is Known when Population Standard Deviation σ is
Unknown
Confidence Intervals for the Population Proportion, p
Determining the Required Sample Size
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Point and Interval Estimates
A point estimate is a single number, a confidence interval provides
additional information about variability
Point Estimate
Lower
Confidence
Limit
Upper
Confidence
Limit
Width of confidence interval
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We can estimate a Population Parameter …
Point Estimates
with a SampleStatistic
(a Point Estimate)
Mean
Proportion pπ
Xμ
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Confidence Intervals
How much uncertainty is associated with a point estimate of a population parameter?
An interval estimate provides more information about a population characteristic than does a point estimate
Such interval estimates are called confidence intervals
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Confidence Interval Estimate
An interval gives a range of values: Takes into consideration variation in
sample statistics from sample to sample Based on observations from 1 sample Gives information about closeness to
unknown population parameters Stated in terms of level of confidence
Can never be 100% confident
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Estimation Process
(mean, μ, is unknown)
Population
Random Sample
Mean X = 50
Sample
I am 95% confident that μ is between 40 & 60.
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General Formula
The general formula for all confidence intervals is:
Point Estimate ± (Critical Value)(Standard Error)
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Confidence Level
Confidence Level
Confidence for which the interval will contain the unknown population parameter
A percentage (less than 100%)
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Confidence Level, (1-)
Suppose confidence level = 95% Also written (1 - ) = 0.95 A relative frequency interpretation:
In the long run, 95% of all the confidence intervals that can be constructed will contain the unknown true parameter
A specific interval either will contain or will not contain the true parameter No probability involved in a specific
interval
(continued)
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Estimates
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Estimates
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Estimates
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Efficient estimator: If the sampling distributions of two statistics have the same mean, the statistic with the smaller variance is called a more efficient estimator of the mean.
Clearly one would in practice prefer to have estimates that are both efficient and unbiased, but this is not always possible.
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Point/Interval EstimatesPoint estimate of the parameter: An estimate of a population parameter given by a single number
Interval estimate of the parameter: An estimate of a population parameter given by two numbers between which the parameter
Reliability: A statement of the error or precision of an estimate
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- A particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate.
- It is an observed interval (i.e. it is calculated from the observations), in principle different from sample to sample, that frequently includes the parameter of interest, if the experiment is repeated.
- How frequently the observed interval contains the parameter is determined by the confidence level or confidence coefficient.
Confidence Interval
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Confidence Interval
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Confidence Intervals
Population Mean
σ Unknown
ConfidenceIntervals
PopulationProportion
σ Known
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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:
where is the point estimate Z is the normal distribution critical value for a probability of /2 in each tail
is the standard error
n
σZX
X
nσ/
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Finding the Critical Value, Z
• Consider a 95% confidence interval:
Z= -1.96 Z= 1.96
0.951
0.0252
α
0.0252
α
Point EstimateLower Confidence Limit
UpperConfidence Limit
Z units:
X units: Point Estimate
0
1.96Z
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Common Levels of Confidence
• Commonly used confidence levels are 90%, 95%, and 99%
Confidence Level
Confidence Coefficient,
Z value
1.28
1.645
1.96
2.33
2.58
3.08
3.27
0.80
0.90
0.95
0.98
0.99
0.998
0.999
80%
90%
95%
98%
99%
99.8%
99.9%
1
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μμx
Intervals and Level of Confidence
Confidence Intervals
Intervals extend from
to
(1-)x100%of intervals constructed contain μ;
()x100% do not.
Sampling Distribution of the Mean
n
σZX
n
σZX
x
x1
x2
/2 /21
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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 0.35 ohms.
• Determine a 95% confidence interval for the true mean resistance of the population.
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2.4068 1.9932
0.2068 2.20
)11(0.35/ 1.96 2.20
n
σZ X
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 0.35 ohms.
• Solution:
(continued)
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Interpretation
• We are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms
• Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean
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Confidence Intervals for Means
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Confidence Intervals for Means
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Confidence Intervals for Means
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Confidence Intervals for Means
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Confidence Intervals
Population Mean
σ Unknown
ConfidenceIntervals
PopulationProportion
σ Known
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If the population standard deviation σ is unknown, we can substitute the sample standard deviation, S
This introduces extra uncertainty, since S is variable from sample to sample
So we use the t distribution instead of the normal distribution
Confidence Interval for μ(σ Unknown)
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Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use large sample
Use Student’s t Distribution 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)
Confidence Interval for μ(σ Unknown)
n
StX 1-n
(continued)
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Student’s t Distribution
The t is a family of distributions The t value depends on degrees of
freedom (d.f.) Number of observations that are free to vary
after sample mean has been calculated
d.f. = n - 1
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If the mean of these three values is 8.0, then X3 must be 9 (i.e., X3 is not free to vary)
Degrees of Freedom (df)
Here, n = 3, so degrees of freedom = n – 1 = 3 – 1 = 2
(2 values can be any numbers, but the third is not free to vary for a given mean)
Idea: Number of observations that are free to vary after sample mean has been calculated
Example: Suppose the mean of 3 numbers is 8.0
Let X1 = 7
Let X2 = 8
What is X3?
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Student’s t Distribution
t0
t (df = 5)
t (df = 13)t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
Standard Normal
(t with df = ∞)
Note: t Z as n increases
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Student’s t Table
Upper Tail Area
df .25 .10 .05
1 1.000 3.078 6.314
2 0.817 1.886 2.920
3 0.765 1.638 2.353
t0 2.920The body of the table contains t values, not probabilities
Let: n = 3 df = n - 1 = 2 = 0.10 /2 = 0.05
/2 = 0.05
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t distribution values
With comparison to the Z value
Confidence t t t Z Level (10 d.f.) (20 d.f.) (30 d.f.) ____
0.80 1.372 1.325 1.310 1.28
0.90 1.812 1.725 1.697 1.645
0.95 2.228 2.086 2.042 1.96
0.99 3.169 2.845 2.750 2.58
Note: t Z as n increases
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Example
A random sample of n = 25 has X = 50 and S = 8. Form a 95% confidence interval for μ
d.f. = n – 1 = 24, so
The confidence interval is
2.0639t0.025,241n,/2 t
25
8(2.0639)50
n
StX 1-n /2,
46.698 ≤ μ ≤ 53.302
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Confidence Intervals for Means
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Confidence Intervals for Means
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Confidence Intervals for Means
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Confidence Intervals for Means
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Confidence Intervals
Population Mean
σ Unknown
ConfidenceIntervals
PopulationProportion
σ Known
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Confidence Intervals for the
Population Proportion, π
An interval estimate for the population proportion ( π ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p )
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Confidence Intervals for the
Population Proportion, π
Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation
We will estimate this with sample data:
(continued)
n
p)p(1
n
)(1σp
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Confidence Interval Endpoints
Upper and lower confidence limits for the population proportion are calculated with the formula
where Z is the standard normal value for the level of confidence
desired p is the sample proportion n is the sample size
n
p)p(1Zp
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Example
A random sample of 100 people shows that 25 are left-handed.
Form a 95% confidence interval for the true proportion of left-handers
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Example A random sample of 100 people
shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers.
/1000.25(0.75)1.9625/100
p)/np(1Zp
0.3349 0.1651
(0.0433) 1.96 0.25
(continued)
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Interpretation
We are 95% confident that the true percentage of left-handers in the population is between
16.51% and 33.49%.
Although the interval from 0.1651 to 0.3349 may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.
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Confidence Intervals for Proportions
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Confidence Intervals for Proportions
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Confidence Intervals for Proportions
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Confidence Intervals for Proportions
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Confidence Intervals for Differences and Sums
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Confidence Intervals for Differences and Sums
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Confidence Intervals for Differences and Sums
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Confidence Intervals for Differences and Sums
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Confidence Intervals for the Variance of a Normal Distribution
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Confidence Intervals for Variance Ratios
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Maximum Likelihood Estimates
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Determining Sample Size
For the Mean
DeterminingSample Size
For theProportion
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Sampling Error
The required sample size can be found to reach a desired margin of error (e) with a specified level of confidence (1 - )
The margin of error is also called sampling error the amount of imprecision in the estimate of the
population parameter the amount added and subtracted to the point
estimate to form the confidence interval
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Determining Sample Size
For the Mean
DeterminingSample Size
n
σZX
n
σZe
Sampling error (margin of error)
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Determining Sample Size
For the Mean
DeterminingSample Size
n
σZe
(continued)
2
22
e
σZn Now solve
for n to get
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Determining Sample Size
To determine the required sample size for the mean, you must know:
The desired level of confidence (1 - ), which determines the critical Z value
The acceptable sampling error, e
The standard deviation, σ
(continued)
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Required Sample Size Example
If = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence?
(Always round up)
219.195
(45)(1.645)
e
σZn
2
22
2
22
So the required sample size is n = 220
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If σ is unknown
If unknown, σ can be estimated when using the required sample size formula
Use a value for σ that is expected to be at least as large as the true σ
Select a pilot sample and estimate σ with the sample standard deviation, S
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Determining Sample Size
DeterminingSample Size
For theProportion
2
2
e
)(1Zn
ππ Now solve
for n to getn
)(1Ze
ππ
(continued)
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Determining Sample Size
To determine the required sample size for the proportion, you must know:
The desired level of confidence (1 - ), which determines the critical Z value
The acceptable sampling error, e
The true proportion of “successes”, π
π can be estimated with a pilot sample, if necessary (or conservatively use π = 0.5)
(continued)
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Required Sample Size Example
How large a sample would be necessary to estimate the true proportion defective in a large population within ±3%, with 95% confidence?
(Assume a pilot sample yields p = 0.12)
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Required Sample Size Example
Solution:
For 95% confidence, use Z = 1.96
e = 0.03
p = 0.12, so use this to estimate π
So use n = 451
450.74(0.03)
0.12)(0.12)(1(1.96)
e
)(1Zn
2
2
2
2
ππ
(continued)
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Ethical Issues
A confidence interval estimate (reflecting sampling error) should always be included when reporting a point estimate
The level of confidence should always be reported
The sample size should be reported An interpretation of the confidence
interval estimate should also be provided