Post on 05-Aug-2020
1/20/03 T-F-test_Examples Steve Brainerd 1
OHSU OGI Class Distributions
ECE-580-DOE : Statistical Process Control and Design of ExperimentsSteve Brainerd
• Statistical Distribution: Types: Continuous
• A statistical distribution which the variables may take on a continuous
range of values. There are over 61 continuous distributions!
• We will use only: Normal Distribution, Student's t-Distribution, and F-Distribution
• Will briefly mention : Chi-Squared Distribution and Weibull Distribution
• Practical applications: Descriptive Data analysis, Population comparisons, SPC, and Design of experiments
• EXCEL: NORMDIST(x,µ,σ,TRUE or FALSE);
• FDIST(x,df1,df2)
• TDIST(x,df,1 or 2 tail)
1/20/03 T-F-test_Examples Steve Brainerd 2
OHSU OGI Class Distributions
ECE-580-DOE : Statistical Process Control and Design of ExperimentsStudent's t-Distribution
• EXCEL function:• TDIST(x,degrees_freedom,tails)• X is the numeric value at which to evaluate the distribution.• Degrees_freedom is an integer indicating the number of degrees of freedom.• Tails specifies the number of distribution tails to return. If tails = 1, TDIST
returns the one-tailed distribution. If tails = 2, TDIST returns the two-tailed distribution..
• TDIST is calculated as TDIST = p( x<X ), where X is a random variable that follows the t-distribution.
• Examples• Returns the Student's t-distribution. The t-distribution is used in the hypothesis
testing of small sample data sets. Use this function in place of a table of critical values for the t-distribution.
• TDIST(1.96,60,2) equals 0.054645• TINV(significance level, degrees of freedom)• TINV(0.05,9) = 2.26 ( two tail at 0.05 or 0.025 one tail)
1/20/03 T-F-test_Examples Steve Brainerd 3
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example
• t distribution table; of values 1 tail/2tail use to check calculation etc in EXCEL
See Z score
1/20/03 T-F-test_Examples Steve Brainerd 4
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example
Normal Probability Plot - Example T-test n = 14
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10Parameter
NO
RM
SIN
V
Sample A
Sample B
99.9%99.4%
97.7%
93.2%84.1%
69.2%50%
30.9%
15.9%6.7%
2.3%0.6%
0.1%
1/20/03 T-F-test_Examples Steve Brainerd 5
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example
Sample A Vs Sample B
0
0.5
1
1.5
2
2.5
3
3.5
7 7.2 7.4 7.6 7.8 8 8.2 8.4
Values
Freq
uenc
ySample ASample B
1/20/03 T-F-test_Examples Steve Brainerd 6
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example# Sample A Sample B t-Test: Two-Sample Assuming Equal Variances1 7.5 7.92 8.2 8.4 Sample A Sample B3 8.1 8 Mean 7.70 7.674 8.4 7.8 Variance 0.18 0.175 7.1 7.5 Observations 15.00 15.006 7.3 7.2 Pooled Variance 0.177 7.1 7.4 Hypothesized Mean Difference 0.008 7.8 7.3 df 28.00
9 8 8.1 t Stat 0.22
10 7.3 7.6 P(T<=t) one-tail 0.41
% Prob of wrongly rejecting Null
11 7.3 7.7 t Critical one-tail 1.70
12 7.9 8.1 P(T<=t) two-tail 0.83
% Prob of wrongly rejecting Null
13 7.8 7.7 t Critical two-tail 2.0514 8.1 6.815 7.6 7.5
mean 7.7 7.6667s 0.42 0.41
Variance 0.18 0.17df 14 14
1/20/03 T-F-test_Examples Steve Brainerd 7
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
F-DistributionFisher F distributionDistribution of the ratio of the standard deviation of n1 randomly
picked numbers by the standard deviation of n2 randomly picked numbers follows an F distribution with n1 –1 and n2-1 degrees of freedom . Note larger standard deviation in numerator!
1/20/03 T-F-test_Examples Steve Brainerd 8
OHSU OGI Class Distributions
ECE-580-DOE : Statistical Process Control and Design of Experiments
F-Distribution• EXCEL function:• FDIST(x,degrees_freedom1,degrees_freedom2)• X is the value at which to evaluate the function.• Degrees_freedom1 is the numerator degrees of freedom.• Degrees_freedom2 is the denominator degrees of freedom.• FDIST is calculated as FDIST=P( F<x ), where F is a random
variable that has an F distribution.• Examples• Returns the F probability distribution. You can use this function to determine
whether two data sets have different degrees of diversity. For example, you can examine test scores given to men and women entering high school and determine if the variability in the females is different from that found in the males.
• FDIST(15.20675,6,4) equals 0.01
1/20/03 T-F-test_Examples Steve Brainerd 9
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Fisher F Distribution
Fisher F distribution used to compare variances from two populations
1/20/03 T-F-test_Examples Steve Brainerd 10
OHSU OGI Class Distributions
ECE-580-DOE : Statistical Process Control and Design of Experiments
F-Distribution• EXCEL functions
1/20/03 T-F-test_Examples Steve Brainerd 11
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Fisher F Distribution
Fisher F distribution
1/20/03 T-F-test_Examples Steve Brainerd 12
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Fisher F distribution example# Sample A Sample B1 7.5 7.92 8.2 8.4 F-Test Two-Sample for Variances3 8.1 84 8.4 7.8 Sample A Sample B5 7.1 7.5 Mean 7.7 7.6666666676 7.3 7.2 Variance 0.176 0.1677 7.1 7.4 Observations 15 158 7.8 7.3 df 14 14
9 8 8.1 F 1.0510 7.3 7.6 P(F<=f) one-tail 0.4611 7.3 7.7 F Critical one-tail 2.4812 7.9 8.113 7.8 7.714 8.1 6.815 7.6 7.5
MANUAL
s 0.4192 0.4082 F cal 1.054285714
Variance 0.1757 0.1667 FINV 2.483723449df 14 14 If Fcal > FINV signifcant difference
1/20/03 T-F-test_Examples Steve Brainerd 13
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example2 Industrial DataProbe card Failures
ProbeCard "A" - Once a failure occurs in testingwafers, card will be return for a complete servicing. (Cleaned,repair, and have card tested on metrology tool before releasing it back for testing)
NOTE: Probecard "A" had a lot more touchdowns then Probecard "B" before it failed again
ProbeCard "B" - Once a failure occurs in testingwafers, card will be examined and brushed cleaned onlyand release back for testing
NOTE: Probecard "B" had a lot less touchdowns before it failed again
Want to compare Probe card A Vs B. Are they the same?
1/20/03 T-F-test_Examples Steve Brainerd 14
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example2 Industrial Data
Testing Wafers with a Serviced Vs Un-service ProbeCard
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 20 40 60 80 100 120 140 160 180 200 220 240
# of Wafers Touched before Failure
Z Sc
ore
Serviced probecard A
Unserviced probecard B
1/20/03 T-F-test_Examples Steve Brainerd 15
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example2 Industrial Data
ProbeCard A Vs Probe card B
0
2
4
6
8
10
12
14
16
18
20
0 30 60 90 120
150
180
210
More
# wafers before failure
Freq
uenc
y
Probe Card AProbe Card A
1/20/03 T-F-test_Examples Steve Brainerd 16
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example2 Industrial Data
Null Hypothesis: mean of # wafers to failure for Probe Card A = mean of # wafers to failure for Probe Card B
Alternative Hypothesis: mean of # wafers to failure for Probe Card A no equal to mean of # wafers to failure for Probe Card B
Ho : µA = µB Hi: µA = µB
We can see from Normal Probability plots that they are very different:How different? Lets go back to this P-value.
It is the risk or chance of wrongly rejecting the null hypothesis of equal means.
I say there is a difference,when there really is not!
1/20/03 T-F-test_Examples Steve Brainerd 17
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example2 Industrial Data
Descriptive Statistics:
Statistic
Serviced ProbeCard "A" (Numbers of Wafers Touched)
UnServiced ProbeCard "B"
(Number of Wafers Touched)
Mean 175.44 74.04Standard Error 3.917012606 4.348113361Median 184 74Mode 184 69Standard Deviation 27.69746176 30.74580443Sample Variance 767.1493878 945.3044898Kurtosis 1.789183309 -1.0973246Skewness -1.419850935 -0.20676057Range 122 115Minimum 89 9Maximum 211 124Sum 8772 3702Count 50 50
1/20/03 T-F-test_Examples Steve Brainerd 18
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example2 Industrial Data
• F test of variances : Accept Null!
F-Test Two-Sample for Variances
UnServiced ProbeCard "B" (Number of Wafers Touched)
Serviced ProbeCard "A" (Numbers of Wafers
Touched)Mean 74.04 175.44
Variance 945.30 767.15Observations 50 50df 49 49
F 1.232
P(F<=f) one-tail 0.234% Prob of wrongly rejecting Null
F Critical one-tail 1.607 5% significance level
1/20/03 T-F-test_Examples Steve Brainerd 19
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example2 Industrial Data
• t test of means : reject Null!t-Test: Two-Sample Assuming Equal Variances
Serviced ProbeCard "A" (Numbers of Wafers Touched)
UnServiced ProbeCard "B" (Number of Wafers Touched)
Mean 175.44 74.04Variance 767 945Observations 50 50Pooled Variance 856
Hypothesized Mean Difference 0df 98
t Stat 17.327P(T<=t) one-tail 6.73966E-32
% Prob of wrongly rejecting Null
t Critical one-tail 1.661 5% significance level
P(T<=t) two-tail 1.34793E-31% Prob of wrongly rejecting Null
t Critical two-tail 1.984 5% significance level
1/20/03 T-F-test_Examples Steve Brainerd 20
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example2 Industrial Data
• t test of means : reject Null!
ProbeCard A Vs Probe card B
0
2
4
6
8
10
12
14
16
18
20
0 30 60 90 120
150
180
210
More
# wafers before failure
Freq
uenc
y
Probe Card AProbe Card A
Statistic ValuePooled Variance 857Polled Std dev 29.27456
n1 50n2 50
Delta means (175.44 - 74) 101.44
Ratio delta/Sp 3.465124
Put in sample size factorsqrt(1/n1 + 1/n2) 0.2
t statistic 17.32562
1/20/03 T-F-test_Examples Steve Brainerd 21
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example2 Industrial Data
• t test of means : Confidence Intervals (note overlap)
Probe Card A Probe card B
mean 175.44 mean 74
std 27.6 std 30.75A
LOWER A UPPER B LOWER B UPPER
z 95% 1.96 121.34 229.54 z 95% 1.96 13.73 134.27
z 99% 2.326 111.24 239.64 z 99% 2.326 2.48 145.52
1/20/03 T-F-test_Examples Steve Brainerd 22
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example3 Industrial Data
• DATA set 1: Large Sample size
Statistic Process A Process B
Mean 0.03198125 -0.017805Standard Error 0.000896024 0.002035238Median 0.03165 -0.0228Mode 0.0079 -0.0402
Standard Deviation 0.018965263 0.022663451
Sample Variance 0.000359681 0.000513632Kurtosis -0.944427736 -0.601604285Skewness 0.080227949 0.3700472Range 0.0734 0.0977Minimum -0.0028 -0.0597Maximum 0.0706 0.038Sum 14.3276 -2.2078Count 448 124
1/20/03 T-F-test_Examples Steve Brainerd 23
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example3 Industrial Data
• DATA set 1Process A Vs B
0
10
20
30
40
50
60
Value
-0.05
5-0.
045
-0.03
5-0.
025
-0.01
5-0.
005
0.005
0.015
0.025
0.035
0.045
0.055
0.065
0.075
Registration Error
Freq
uenc
y
Process AProcess B
1/20/03 T-F-test_Examples Steve Brainerd 24
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example3 Industrial Data
• DATA set 1: Means are Significantly Different
F-Test Two-Sample for Variances
t-Test: Two-Sample Assuming Unequal
Variances
Process B Process A Process A Process BMean -0.01780 0.03198 Mean 0.03198 -0.01780
Variance 0.00051 0.00036 Variance 0.00036 0.00051Observations 124.00000 448.00000 Observations 448.00000 124.00000
df 123.00000 447.00000Hypothesized Mean
Difference 0.00000F 1.42802 df 174.00000
P(F<=f) one-tail 0.00494 t Stat 22.38837F Critical one-tail 1.25647 P(T<=t) one-tail 0.00000
t Critical one-tail 1.65366P(T<=t) two-tail 0.00000
t Critical two-tail 1.97369
1/20/03 T-F-test_Examples Steve Brainerd 25
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example4 Industrial Data
• DATA set 2: Large Sample size
Normal Probability Plot - dat set 2
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-0.1 -0.05 0 0.05 0.1
Parameter
NO
RM
SIN
V
Process A
New Process B
99.9%
99.4%
97.7%
93.2%
84.1%
69.2%
50%
30.9%
15.9%
6.7%
2.3%
0.6%
0.1%
1/20/03 T-F-test_Examples Steve Brainerd 26
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example4 Industrial Data
• DATA set 2: Means are Significantly DifferentF-Test Two-Sample for Variances Large Sample
t-Test: Two-Sample Assuming Unequal
Variances Large Sample
New Process B large sample
Process A large sample
Process A large sample
Mean 0.00220 0.03198 Mean 0.03198Variance 0.00051 0.00036 Variance 0.00036Observations 124.00000 448.00000 Observations 448.00000
df 123.00000 447.00000Hypothesized Mean Difference 0.00000
F 1.42802 df 174.00000P(F<=f) one-tail 0.00494 t Stat 13.39455
F Critical one-tail 1.25647 P(T<=t) one-tail 0.00000t Critical one-tail 1.65366
P(T<=t) two-tail 0.00000t Critical two-tail 1.97369
1/20/03 T-F-test_Examples Steve Brainerd 27
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example4a Industrial Data
• DATA set 2: had 174 degrees of freedom for t-test:• What happens if we reduce the sample size to 18?
Normal Probability Plot
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-0.1 -0.05 0 0.05 0.1
Parameter
NO
RM
SIN
V
Process A
New Process B small sample
99.9%99.4%
97.7%
93.2%84.1%
69.2%50%
30.9%
15.9%6.7%
2.3%0.6%
0.1%
Small Sample A Vs B
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
1 2 3 4 5 6 7 8 9 10 11
Value
Freq
uenc
y
Process A
Process B
1/20/03 T-F-test_Examples Steve Brainerd 28
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example4a Industrial Data
• DATA set 2: Means are now just barely different F-Test Two-Sample for Variances t-Test: Two-Sample Assuming Equal Variances
New Process B small sample
Process A small sample
Process A small sample
New Process B small sample
Mean 0.014622222 0.032672222 Mean 0.032672222 0.014622222Variance 0.000565403 0.000342246 Variance 0.000342246 0.000565403
Observations 18 18 Observations 18 18
df 17 17 Pooled Variance 0.000453824
F 1.65203853Hypothesized
Mean Difference 0
P(F<=f) one-tail 0.155143353% Prob of wrongly rejecting Null df 34
F Critical one-tail 2.271892896 t Stat 2.54187725
P(T<=t) one-tail 0.00787627% Prob of wrongly rejecting Null
t Critical one-tail 1.690923455 5% significance level
P(T<=t) two-tail 0.015752539% Prob of wrongly rejecting Null
t Critical two-tail 2.032243174 5% significance level
1/20/03 T-F-test_Examples Steve Brainerd 29
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example5 Industrial Data
• DATA set 3 Large Sample size
Normal Probability Plot Dat set 3
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-0.1 -0.05 0 0.05 0.1
Parameter
NO
RM
SIN
V
Process A
New Process B rev 3
99.9%99.4%
97.7%
93.2%84.1%
69.2%50%
30.9%
15.9%6.7%
2.3%0.6%
0.1%
1/20/03 T-F-test_Examples Steve Brainerd 30
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example5 Industrial Data
• DATA set 3: Means are Significantly Different
F-Test Two-Sample for Variances
t-Test: Two-Sample Assuming Unequal Variances
New Process B rev 3 Process A Process A
New Process B rev 3
Mean 0.02220 0.03198 Mean 0.03198 0.02220Variance 0.00051 0.00036 Variance 0.00036 0.00051Observations 124.00000 448.00000 Observations 448.00000 124.00000
df 123.00000 447.00000Hypothesized Mean Difference 0.00000
F 1.42802 df 174.00000P(F<=f) one-tail 0.00494 t Stat 4.40072F Critical one-tail 1.25647 P(T<=t) one-tail 0.00001
t Critical one-tail 1.65366P(T<=t) two-tail 0.00002t Critical two-tail 1.97369
1/20/03 T-F-test_Examples Steve Brainerd 31
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example6 Industrial Data
• DATA set 4: Large Sample size
1/20/03 T-F-test_Examples Steve Brainerd 32
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example6 Industrial Data
• DATA set 4 : Means are Significantly Different Remember this is a comparison of means!
F-Test Two-Sample for Variances
t-Test: Two-Sample Assuming Unequal Variances
New Process B rev 4 Process A Process A
New Process B rev 4
Mean 0.02720 0.03198 Mean 0.03198 0.02720Variance 0.00051 0.00036 Variance 0.00036 0.00051Observations 124.00000 448.00000 Observations 448.00000 124.00000
df 123.00000 447.00000Hypothesized Mean Difference 0.00000
F 1.42802 df 174.00000P(F<=f) one-tail 0.00494 t Stat 2.15226
F Critical one-tail 1.25647 P(T<=t) one-tail 0.01638t Critical one-tail 1.65366P(T<=t) two-tail 0.03275t Critical two-tail 1.97369
1/20/03 T-F-test_Examples Steve Brainerd 33
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example7 Industrial Data
• DATA set 5: Large Sample size
1/20/03 T-F-test_Examples Steve Brainerd 34
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example7 Industrial Data
• DATA set 5: No significant difference between means found
F-Test Two-Sample for Variances
t-Test: Two-Sample Assuming Unequal Variances
New Process B rev 2 Process A Process A
New Process B rev 2
Mean 0.02920 0.03198 Mean 0.03198 0.02920Variance 0.00051 0.00036 Variance 0.00036 0.00051Observations 124.00000 448.00000 Observations 448.00000 124.00000
df 123.00000 447.00000Hypothesized Mean Difference 0.00000
F 1.42802 df 174.00000P(F<=f) one-tail 0.00494 t Stat 1.25288F Critical one-tail 1.25647 P(T<=t) one-tail 0.10597
t Critical one-tail 1.65366P(T<=t) two-tail 0.21193t Critical two-tail 1.97369
1/20/03 T-F-test_Examples Steve Brainerd 35
OHSU OGI Class ECE-580-DOE :Statistical Process Control and Design of Experiments Steve Brainerd
Basic Statistics Student t Distribution Example8 Industrial Data
• DATA set 6 if both Samples are Identical t-Test: Two-Sample Assuming Equal Variances
Process A Process BMean 0.03198 0.03198
Variance 0.00036 0.00036Observations 448.00000 448.00000
Pooled Variance 0.00036Hypothesized Mean Difference 0.00000
df 894.00t Stat 0.00
P(T<=t) one-tail 0.50% Prob of wrongly rejecting Null
t Critical one-tail 1.65 5% significance level
P(T<=t) two-tail 1.00% Prob of wrongly rejecting Null
t Critical two-tail 1.96 5% significance level