Construction Engineering 221

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Construction Engineering 221

• Normal distribution part II

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Normal distribution

• Many measurements are normally distributed and
  can be converted to a standardized distribution
  in order to use z-tables
• To calculate probabilities, it is typical to pick
  an acceptable level (say, .05) and calculate
  probability that z is greater than or less than
  that value (this will be important in hypothesis
  testing)

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Normal distribution

• Example
• Concrete cylinder test results (in psi) are 1000,
  1200, 1150, 1100, and 1230. You need to have
  three of six cylinders exceed 1200 psi before you
  can strip forms. What is the probability that
  the sixth cylinder will exceed 1200 psi?
• First calculate the mean and sd of the sample
• Mean 1136, sd 90

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Normal distribution

• Z-statistic for the sixth cylinder is
• 1200-1136/90 .711
• Z .711, P .26115
• Probability is (1 - .76115), or .2389, 24 chance
• must add 50 area from left of the mean and
  subtract remaining area to find probability

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Normal distribution

• What if the specifications read that the average
  of six cylinders must be 1200?
• 1200 X 6 7200
• S (1-5) 5680
• Cylinder 6 must test at 7200-5680 1520 for
  average to be 1200
• Z 1520-1200/176.5 1.81
• A(x) .46485
• P 1-(.5 .46485) .0352, or 3 chance

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Normal distribution

• What if you asked to change the specification to
  read the average of the 5 highest of 6 tests
  (eliminate the outlier of 1000) must equal 1200?
• 1200 1150 1100 1230 X/5 1200
• X 1320 (sixth cylinder must test at 1320
• Z 1320-1200/83.37 1.44
• A(x) .42507
• P 1-.92507, or 7.5

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Normal Distribution

• Is this important?
• Assume an 18 million dollar job with a 12 month
  completion schedule
• 1.5 million per month/ 22 workdays per 68,200
  per day
• .03 X 68200 2046
• .075 X 68200 5115
• risk savings of over 3000 for a simple change
• Consider you make (50,000 X1.4)/264 or 265/day

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Normal distribution

• Normal approximation of the Binomial distribution
  is used when the number of trials is large
• Math gets cumbersome
• Large sample sizes approach continuous
  distributions
• Use normal approximations when values exceed
  those included in binomial tables
• Use when np gt5 (or n (1-p)), or when ngt30

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Normal distribution

• Do 6 and 10 on page 82