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For every x 1. 0 ≤ P(x) ≤ 1 2. ∑P(x)=1 #defects01234 p(x) ?

#defects01234 p(x) Mean = St. Dev = (Expected Value) (L ) 2 x L 2 Calculator

#defects01234 p(x)

Linear Function: Mean of Variance of

Properties: 1.Fixed # of trials 2. Each trial results in 1 of 2 outcomes 3. Outcomes of different trials are independent 4. P(x) remains the same

Y N Y N Y N Y N Y Y N N Y N Prob. Dist of Yes’s on survey YYYNYY YYNNYN NYYNNY NYNNNN

# yes’sP(x) *remember Pascal’s triangle? Calculator:P(x=2): binompdf(3,.6, 2) = P(x<2): binomcdf(3,.6, 1) = P(x>2): 1 - binomcdf(3,.5, 2) = 0.216

Mean: St. Dev: Given: total number – n Probability Ask: Exactly x At least x More than x

Example: The probability that a certain machine will produce a defective item is If a random sample of 6 items is taken from the output of this machine, what is the probability that there will be more than 4 defective items in this sample? 1-binomcdf(6,0.2, 4) = At least 2 items? 1-binomcdf(6, 0.2, 1)=

“Go Until” The probability that a person will wear glasses is What is the probability that exactly 3 people must be picked before one with glasses is included in the sample? Calculator: Geompdf(0.3, 3)

Standard Normal: (called the z – curve) Calculator: Use normalcdf (beginning, end, mean, st. dev) Use to Find Probability To Find a Value Calculator: Use invnorm (%, mean, st. dev)

Example: IQ scores are normally distributed with a mean of 100 and a st. dev. of 15. What is the probability that you select a student at random with an IQ greater than 120? What is the IQ score associated with the 25 th percentile? Normcdf (120,∞,100,15) = 0.09 Invnorm (.25, 100,15)= 89

Example: Find the mean and standard deviation of normally distributed data if 27.7% is less than 18 and 33.3% is greater than

Properties for Mean: If population is normal – then sampling distribution is approx. normal 4. Central Limit Theorem – if n is large enough, its approx normal Properties for Proportion: Approx. normal if: An n increasesdecreases