Statistical Analysis of Experimental data

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Statistical Analysis of Experimental data
Feb. 15, 2007
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Experimental Errors

• Systematic error reduce by calibration
• Random error statistical analysis

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General Definitions

• Population
• The entire collection of objects, measurements,
  etc.
• Sample
• A representative subset of a population
• Sample Space
• The set of possible outcomes of an experiment
  (discrete, continuous)
• Random Variable
• The variable being measured
• Probability
• The chance of the occurrence of an event in an
  experiment

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Measures of Central Tendency

• Mean (sample mean, population mean)
• Median
• Ascending or descending order
• Odd middle one
• Even avg. of the middle two
• Mode
• Highest frequency of occurrence

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Measures of Dispersion

• Dispersion
• Spread or variability of the data
• Deviation
• Mean deviation
• Standard deviation (Sample, Population)
• Variance

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Basics of Probability

• Probability
• successful occurrences
• Probability of event A
• total number of possible outcomes
• If A is certain to occur, P(A) 1
• If A is certain not to occur, P(A) 0
• If B is the complement of A, then P(B) 1-P(A)
• If A and B are mutually exclusive (the
  probability of simultaneous occurrence is zero),
  P(A or B) P(A) P(B)
• If A and B are independent, P(AB) P(A)P(B)
  (occurrence of both A and B)
• P(A?B) P(A)P(B)-P(AB) (Occurrence of A or B or
  both)

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Basics of Probability

• Probability Mass Function
• Normalization
• Mean
• Variance

N 3 Hit purple region, score 3 Hit blue
region, score 2 Hit red region, score 1
P(p)ltP(b)ltP(r)
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Basics of Probability (continued)

• Probability Density Function
• Probability of occurrence in an interval xi and
  xidx
• Probability of occurrence in an interval a,b
• Mean of population
• Variance of population

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Probability distribution Function? Normal
Distribution

• Symmetric about m
• Bell-shaped
• Mean m the peak of the density occurs
• Standard deviation s indicates the spread of the
  bell curve.

m 2
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Standard Normal Distribution (mean0, standard
deviation1)
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Normal Distribution Example

• The distribution of heights of American women
  aged 18 to 24 is approximately normally
  distributed with mean 65.5 inches and standard
  deviation 2.5 inches.
• 68 of these American women have heights between
  65.5 1(2.5) and 65.5 1(2.5) inches, or
  between 63 and 68 inches,
• 95 of these American women have heights between
  65.5 - 2(2.5) and 65.5 2(2.5) inches, or
  between 60.5 and 70.5 inches.
• 99.7 of these American women have heights
  between 65.5 - 3(2.5) and 65.5 3(2.5) inches,
  or between 58 and 73 inches.

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Parameter Estimation

• Estimate population mean using sample mean
• Estimate population standard deviation using
  sample standard deviation

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Interval Estimation of Population Mean

• Estimate of population mean
• Confidence interval
• Confidence level (degree of confidence)
• Level of significance
• a 1 confidence level

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From Sample to Population

• Choose many samples from population
• Find the mean of each sample
• Determine the uncertainly range of the means
• (Sample mean is a variable !!!)
• This method is very costly
• How to estimate the statistics of a population
  from only one single sample?

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Central Limit Theorem (a??)

• Several different samples, each of size n, mean
  of
• If sample size n is sufficiently large (gt30),
  then
• 
  (standard error of the mean)
• If original population is normal, the
  distribution for the is normal
• If original population is not normal and n is
  large (n gt 30) the distribution for the is
  normal
• If original population is not normal and n lt 30
  the distribution for the is only
  approximately normal

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Interval Estimation of Population Mean (ngt30)

With confidence level 1-a
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Normal Distribution Table
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Example 6.11 (P141)

• n 36
• Average 25 ?
• Sample standard deviation S 0.5 ?
• Determine 90 confidence interval of the mean
• Solution
• 1-a90, -gt a 0.1
• 0.5- a/20.5-0.050.45
• Table 6.3 -gt Z a/21.645

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Students t-distribution (nlt30)

• Students t, degree of freedom ? n-1

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Interval Estimation of Population Mean (nlt30)

With confidence level 1-a
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Srudents t-Distribution Table
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Example 6.12 (P145)

• n6
• 1250, 1320, 1542, 1464, 1275, and 1383 h
• Estimate population mean and 95 confidence
  interval on the mean
• Solution
• Mean(125013201542146412751383)/61372 h
• S(1250-1372)2(1320-1372)2 (1542-1372)2
  (1464-1372)2 (1275-1372)2(1383-1372)2/(6-1)1/2
  114 h
• ?n-16-15, a1-950.05, a/20.05/20.025
• Table 6.6 -gt t a/22.571

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Example 6.13 (P145)

• Reduce the 95 confidence interval to 50 h from
  120 h
• Determine how many more systems should be tested
• Solution Assume ngt30,
• a1-950.05, 0.5-a/20.5-0.05/20.475
• Table 6.3 -gt z a/21.96
• n (1.96x114/50)220 lt30 gt
  t-distribution
• ?n-120-119, a/20.05/20.025
• Table 6.6 -gt t a/22.093
• n (2.093x114/50)223

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Kye-squared Distribution Function

• Relates the sample variance to the population
  variance

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Interval Estimation of Population Variance

With confidence level 1-a
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Chi-squre Distribution Table
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Example 6.14 (P149)

• n 20, mean 0.32500 in, S 0.00010 in
• obtain 95 confidence interval for the standard
  deviation
• Solution
• ?n-120-119, a1-950.05, a/20.05/20.025,
  1- a/21-0.05/20.975
• Table 6.7 -gt ?2?, a/232.825, ?2?, 1-a/28.9065

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Assignment

• Reading
• Ch 6.3 (P133-138), 6.4
• Homework
• 6.46, 6.47, 6.49, 6.52, 6.53