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4.3 NORMAL PROBABILITY DISTRIBUTIONS The Most Important Probability Distribution in Statistics

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A random variable X with mean and standard deviation is normally distributed if its probability density function is given by Normal Distributions

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The Shape of Normal Distributions Normal distributions are bell shaped, and symmetrical around Why symmetrical? Let = 100. Suppose x = 110. Now suppose x =

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Normal Probability Distributions The expected value (also called the mean) E(X) (or ) can be any number n The standard deviation can be any nonnegative number n The total area under every normal curve is 1 n There are infinitely many normal distributions

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Total area =1; symmetric around µ

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The effects of and How does the standard deviation affect the shape of f(x)? = 2 =3 =4 = 10 = 11 = 12 How does the expected value affect the location of f(x)?

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X A family of bell-shaped curves that differ only in their means and standard deviations. µ = the mean of the distribution = the standard deviation µ = 3 and = 1

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X X µ = 3 and = 1 µ = 6 and = 1

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X X µ = 6 and = 2 µ = 6 and = 1

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X Probability = area under the density curve P(6 < X < 8) = area under the density curve between 6 and 8. ab b a P(6 < X < 8) µ = 6 and = 2 0 X

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X Probability = area under the density curve P(6 < X < 8) = area under the density curve between 6 and 8. ab b a P(6 < X < 8) µ = 6 and = X

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X Probability = area under the density curve P(6 < X < 8) = area under the density curve between 6 and 8. ab b a 68 68

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ab Probabilities: area under graph of f(x) P(a < X < b) = area under the density curve between a and b. P(X=a) = 0 P(a < x < b) = P(a < x < b) f(x) P(a < X < b) X

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Standardizing Suppose X~N( Form a new random variable by subtracting the mean from X and dividing by the standard deviation : (X n This process is called standardizing the random variable X.

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Standardizing (cont.) (X is also a normal random variable; we will denote it by Z: Z = (X has mean 0 and standard deviation 1: E(Z) = = 0; SD(Z) = n The probability distribution of Z is called the standard normal distribution.

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Standardizing (cont.) n If X has mean and stand. dev. , standardizing a particular value of x tells how many standard deviations x is above or below the mean . n Exam 1: =80, =10; exam 1 score: 92 Exam 2: =80, =8; exam 2 score: 90 Which score is better?

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X µ = 6 and = 2 Z µ = 0 and = 1 (X-6)/2

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Pdf of a standard normal rv n A normal random variable x has the following pdf:

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Z = standard normal random variable = 0 and = 1 Z Standard Normal Distribution.5

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Important Properties of Z #1. The standard normal curve is symmetric around the mean 0 #2.The total area under the curve is 1; so (from #1) the area to the left of 0 is 1/2, and the area to the right of 0 is 1/2

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Finding Normal Percentiles by Hand (cont.) n Table Z is the standard Normal table. We have to convert our data to z-scores before using the table. n The figure shows us how to find the area to the left when we have a z-score of 1.80:

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Areas Under the Z Curve: Using the Table P(0 < Z < 1) = = Z

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Standard normal probabilities have been calculated and are provided in table Z. The tabulated probabilities correspond to the area between Z= - and some z 0 Z = z 0 P(-

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n Example – continued X~N(60, 8) In this example z 0 = = z P(z < 1.25)

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Examples n P(0 z 1.27) = z Area= =.3980

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P(Z .55) = A 1 = 1 - A 2 = = A2A2

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Examples n P(-2.24 z 0) = Area= =.4875 z Area=.0125

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P(z -1.85) =.0322

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Examples (cont.) A1A1 A2A z n P(-1.18 z 2.73)= A - A 1 n = n = A1A1 A

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P(-1 ≤ Z ≤ 1) = =.6826 vi) P(-1≤ Z ≤ 1)

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Is k positive or negative? Direction of inequality; magnitude of probability Look up.2514 in body of table; corresponding entry is P(z < k) =

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Examples (cont.) viii)

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Examples (cont.) ix)

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X~N(275, 43) find k so that P(x

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P( Z < 2.16) = Z Area=

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Example Regulate blue dye for mixing paint; machine can be set to discharge an average of ml./can of paint. Amount discharged: N( ,.4 ml). If more than 6 ml. discharged into paint can, shade of blue is unacceptable. Determine the setting so that only 1% of the cans of paint will be unacceptable

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Solution

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Solution (cont.)

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