statistical distribution: types:...

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)



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)


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




Normal Probability Plot - Example T-test n = 14

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0 2 4 6 8 10Parameter

NO

RM

SIN

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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%




Sample A Vs Sample B

0

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7 7.2 7.4 7.6 7.8 8 8.2 8.4

Values

Freq

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ySample ASample B



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


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



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!



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



Basic Statistics Fisher F Distribution

Fisher F distribution used to compare variances from two populations



ECE-580-DOE : Statistical Process Control and Design of Experiments

F-Distribution• EXCEL functions



Basic Statistics Fisher F Distribution

Fisher F distribution



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



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?



Basic Statistics Student t Distribution Example2 Industrial Data

Testing Wafers with a Serviced Vs Un-service ProbeCard

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# of Wafers Touched before Failure

Z Sc

ore

Serviced probecard A

Unserviced probecard B




ProbeCard A Vs Probe card B

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Probe Card AProbe Card A




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!




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




• 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




• 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


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




• t test of means : reject Null!

ProbeCard A Vs Probe card B

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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




• 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




• 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




• DATA set 1Process A Vs B

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Registration Error

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Process AProcess B




• DATA set 1: Means are Significantly Different


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





Normal Probability Plot - dat set 2

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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%




• 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



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

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Parameter

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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

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Process A

Process B



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


Process A small sample

Process A 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


P(T<=t) two-tail 0.015752539% Prob of wrongly rejecting Null





• DATA set 3 Large Sample size

Normal Probability Plot Dat set 3

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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%




• DATA set 3: Means are Significantly Different


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


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




• DATA set 4 : Means are Significantly Different Remember this is a comparison of means!




New Process B rev 4



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




• DATA set 5: No significant difference between means found




New Process B rev 2



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




• 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


P(T<=t) two-tail 1.00% Prob of wrongly rejecting Null


statistical distribution: types:...

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