Download Confidence Intervals

Confidence Intervals
Confidence Interval
• We are trying to draw a conclusion about a
population based on a finite sample
• We can not be 100% sure about our
conclusion
• Instead, we express a confidence interval
• The distribution depends upon the statistical
variable being considered (e.g. Chi-square
distribution for variance)
Consider the Following Table
•
•
•
•
100 values (the population)
Mean, μ = 26.1
Variance, σ² = 17.5
We will take a sample of 10 values from the
population and draw a statistical conclusion
• Our sample mean and variance won’t match
the population exactly, but should be within
a certain tolerance
18.2 26.4 20.1 29.9 29.8 26.6 26.2
25.7 25.2 26.3 26.7 30.6 22.6 22.3
30.0 26.5 28.1 25.6 20.3 35.5 22.9
30.7 32.2 22.2 29.2 26.1 26.8 25.3
24.3 24.4 29.0 25.0 29.9 25.2 20.8
29.0 21.9 25.4 27.3 23.4 38.2 22.6
28.0 24.0 19.4 27.0 32.0 27.3 15.3
26.5 31.5 28.0 22.4 23.4 21.2 27.7
27.1 27.0 25.2 24.0 24.5 23.8 28.2
26.8 27.7 39.8 19.8 29.3 28.5 24.7
22.0 18.4 26.4 24.2 29.9 21.8 36.0
21.3 28.8 22.8 28.5 30.9 19.1 28.1
30.3 26.5 26.9 26.6 28.2 24.2 25.5
30.2 18.9 28.9 27.6 19.6 27.9 24.9
21.3 26.7
Class Experiment
• Randomly select data sets of 10 values
• Compute mean and variance for the sample.
• Compare to population mean and variance
10 Random Sets of 10
Set 1: 24.2 24.4 28.5 25.3 32.2 19.6 32.9 21.3 24.0 26.5
Set 2: 33.9 21.3 21.3 25.2 18.9 19.6 28.5 36.0 27.1 30.6
Set 3: 28.0 21.2 18.9 33.2 30.2 26.5 25.2 29.0 21.8 26.3
Set 4: 32.2 30.0 24.2 18.9 17.2 22.4 21.3 21.3 26.4 24.5
Set 5: 25.4 25.2 21.3 32.2 22.6 21.3 25.7 22.4 23.1 25.3
Set 6: 32.2 28.9 27.0 20.8 20.3 18.4 31.5 26.8 33.2 27.3
Set 7: 22.0 25.3 26.5 32.2 25.4 28.5 22.7 24.2 25.5 27.3
Set 8: 30.3 20.3 20.9 22.8 19.1 23.1 25.3 30.9 19.4 28.0
Set 9: 21.3 25.6 25.8 24.7 28.9 30.2 21.3 25.2 27.9 25.7
Set 10:21.3 32.0 21.3 23.1 30.0 24.0 26.8 29.0 30.6 26.8
Sample Statistics
Mean Variance
Set 1:
Set 2:
Set 3:
Set 4:
Set 5:
Set 6:
Set 7:
Set 8:
Set 9:
Set 10:
25.89
26.24
26.03
23.84
24.45
26.64
25.96
24.01
25.66
26.49
18.42
36.29
19.39
22.05
10.35
27.27
8.65
19.52
8.37
15.29
Note the variation in
sample statistics
(Recall, population
mean and variance are
26.1 and 17.5)
Effect of Increasing Sample Size
Sampling Distribution Theory
• Greater confidence with larger samples
• We use estimators to make inferences about
populations
• Two estimators were already discussed –
sample mean and sample variance
• The estimators themselves are random
variables, each having a particular distribution
Chi-Square
2
(χ )
Distribution
• Compares the relationship between
population variance and sample variance
• Depends on the sample size and therefore
the number of degrees of freedom
• Not symmetric – skewed to the right
2 
vS 2
2
v = degrees of freedom = n - 1
Probability density
Probability, α, is area
under curve
Student t Distribution
• Compares the relationship between
population mean and sample mean
• Also depends on degrees of freedom
• Symmetric
• As degrees of freedom approach infinity, it
approaches a normal distribution
y
t
S
n
Essentially, deviation from the mean
divided by standard deviation of the
mean
Probability, α, is area
under curve
F Distribution
• Compares relationship between ratio of two
population variances and ratio of two sample
variances
• Depends on degrees of freedom of both
samples
• Shape is similar to Chi-square
• Need different tables for different levels of α
S12
F
S 22
 12
 22
S12  22
 2 2
S2  1
v1 = degrees of freedom in numerator
v2 = degrees of freedom in denominator
Also
F1 ,v1 ,v2 
1
F ,v2 ,v1
Probability, α, is area
under curve
Confidence Interval for the Mean
• Normal distribution is used for populations
• For finite samples, use the Student-t
distribution
• Once the sample size reaches about 30, the
distribution becomes approximately normal
(to about 2 significant figures)
• For an interval, divide probability, α, by 2
for correct t-table value


 y

P
 t   1
S

n


y  t / 2
S
S
   y  t / 2
n
n
EXAMPLE
A sample of 20 circle readings has a mean of 34.5", and a standard
deviation of ±2.1", what is the:
a) 95% confidence interval for the pop. mean?
b) 99% confidence interval for the pop. mean?
c) would a measurement of 35.7 be acceptable for this set of data?
Part a)
Step 1: = 0.05 (1 - 0.95) so /2 = 0.025, v = 20 - 1 = 19
Look up critical value of t = 2.093 (0.025, 19)
Step 2:
2.1
2.1
33.5  34.5  2.093 
   34.5  2.093 
 35.5
20
20
Part b: 99% CONFIDENCE INTERVAL
Step 1: = 0.01 (1 - 0.99) so /2 = 0.005, v = 20 - 1 = 19
Look up critical value of t = 2.861 (0.005, 19)
Step 2:
2.1
2.1
33.2  34.5  2.861
   34.5  2.861
 35.8
20
20
Note that the 99% confidence interval is larger than the 95%. This
interval indicates that 99% of the time the population mean is between
33.2 and 35.8.
Part c: A value of 35.7 is marginal. It is outside the 95% confidence
region, but within the 99% confidence region.
Confidence Intervals from Samples
Construct a 90% confidence interval for µ. Does the µ of 26.1
lie in the interval?
SET 1: 23.40 < µ < 28.38
SET 2: 22.75 < µ < 29.73
SET 3: 23.48 < µ < 28.58
SET 4: 21.12 < µ < 26.56
SET 5: 22.59 < µ < 26.31
SET 6: 23.61 < µ < 29.67
SET 7: 24.26 < µ < 27.66
SET 8: 21.45 < µ < 26.57
SET 9: 23.98 < µ < 27.34
SET 10: 24.22 < µ < 28.76
Selecting a Sample Size
Confidence Interval for Variance
P
    / 2,v   1  
2
1 / 2,v
2
2
 2

vS 2
2
P 1 / 2,v  2   / 2,v   1  



Confidence
Interval
vS 2
 / 2,v
2
 
2
vS 2
12 / 2,v
Confidence Intervals from Samples
Confidence Interval for Ratio of
Variances


S12  22
P F1 / 2,v1 ,v2  2  2  F / 2,v1 ,v2   1  
S2  1


After some manipulation (see text)…
The confidence interval is:
S12
1
 12 S12

 2  2  F / 2,v2 ,v1
2
S 2 F / 2,v1 ,v2  2 S 2
EXAMPLE
On Day 1, 10 EDM distance measurements result in a variance of
52 mm2. On Day 2, 21 additional measurements of the same
distance result in a variance of 61 mm2. What is the 95%
confidence interval for the ratio of the population variances? In
similar measurement conditions, the expected ratio of the
variances is 1, i.e., σ12 = σ22. From the constructed interval is this
true?
Confidence Intervals from Samples
Solutions
1. (0.13, 2.05)
2. (0.24, 3.83)
3. (0.21,3.37)
4. (0.44, 7.17)
5. (0.17, 2.72)
6. (0.53, 8.58)
7. (0.23, 3.80)
8. (0.55, 8.87)
9. (0.30, 4.85)
10. (0.49, 7.94) 11.(1.07, 17.5)* 12. (0.30, 4.98)
Note that set 11 does not contain 1. Thus there is reason to believe
that samples 2 and 9 are not from the same population at a 95% level
of confidence. This assumption is obviously wrong, and thus the test
has given an incorrect result, which can be expected 5% of the time.