Lecture2-2

# Lecture2-2 - 2 2 2 1 2 1 = σ μ π σ x e y y 3 Empirical...

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Unformatted text preview: 2 2 2 1 2 1 -- = σ μ π σ x e y y 3 Empirical Rule for Normal Populations 4 5 How to calculate these normal probabilities? Verbal SAT scores for college-bound seniors in 1999 have approximately normal distribution with µ = 505 and σ = 110. What is the proportion of scores below 615? 6 7 Start by drawing a picture 8 Note: 615 is 1 SD above the mean. (615 = 505 + 110) 9 68.26 % 15.87 % 15.87 % Answer = 15.87% + 68.26% = 84.13% 615 505 395 x : an individual observation from a normal distribution with mean μ and standard deviation σ z-score: describes how many SD’s x is above or below the mean 10 σ μ- = x z z-score If X is normally distributed with mean μ and standard deviation σ , then the random variable is normally distributed with mean 0 and standard deviation 1 ( standard normal ). So if x is an observed value from normal with mean μ and SD σ , then the z-score for x is from standard normal . 11 σ μ- = X Z 12 The area under a normal density (mean μ , SD σ ) to the left (or right ) of x = the area under the standard normal density to the left (or right ) of z , where z is the z-score for x For example: 13 σ μ- = x z ) ( ) ( σ μ- < < = < < x Z P x X P 14 Find P (0 ≤ Z ≤ 1) Find the area listed in the table corresponding to z = 1.00 Starting from the top of the far left column, go down to “1.0” Read across the row z = 1.0 until under the column “.00” The area is in the cell that is the intersection of the row and column As listed in the table, the area is 0.3413, so P (0 ≤ Z ≤ 1) = 0.3413 15 16 17 Z: Standard normal random variable 18 19 20 Think of P(X < x) as the probability of an event where “X< x” is the event When dealing with distributions P(X < x) can also be interpreted as a proportion or relative frequency Table gives us P(0< Z < z ) when Z ~ N(0,1) P(Z < -z ) = P(Z > z ), i.e. they are symmetric P(z 1 < Z < z 2 ) = P( 0<Z < z 2 ) - P(0<Z < z 1 ) if z 1 , z 2 >0 P(Z < z ) = P(Z < z ), since P( Z = z ) = 0...
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Lecture2-2 - 2 2 2 1 2 1 = σ μ π σ x e y y 3 Empirical...

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