Relative Frequency Histogram. Density Curve. Continuous Distribution. Continuous Random Variable. Meaning of Area Under Curve. Normal Distribution

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1 Relative Frequency Histogram Percent Moeling Continuous Test scores 1 Density function, f (x) Density Curve Percent A smooth curve that fit the istribution Test scores Use a mathematical moel to escribe the variable. 3 Continuous Density Function (Curve) 1. Mathematical Formula Shows Probability Densities,, for All Values of x, & [ Is Not Probability ]. Property Total Area Uner Curve is 1. Probability Density Function (Value, Density) Value 4 x Continuous Ranom Variable P( c ) f ( x ) x c Meaning of Area Uner Curve : What percentage of the istribution is in between 7 an 6? Probability Is Area Uner Curve! c P(7) (Height) 6-1

2 Uniform Skewe to right Symmetrical Skewe to left 1. Bell-Shape & Symmetrical. Mean, Meian, Moe Are Equal 3. Ranom Variable Has Infinite Range < x < f() Mean Meian Moe 7 Probability Density Curve Stanar 9 Probability Density Function 1 1 x f ( x ) e π f (x ) Density of Ranom Variable x Mean of the Stanar Deviation of the π ; e.71 x Value of Ranom Variable ( < x < ) Notation: N (, ) Α normal istribution with mean an stanar eviation 10 Effect of Varying Parameters ( & ) N (7, ) Α normal istribution with mean 7 an stanar eviation. Possible situations: Test scores, pulse rates, f() B N (130, 4) Α normal istribution with mean 130 an stanar eviation 4. Possible situations: Weight, Cholesterol levels, A C

3 Probability is area uner curve! Probability c P( c x ) f ( x) x Stanar Stanar : A normal istribution with mean 0 an stanar eviation 1. Notation: 1 c x 13 ~ N ( 0, 1) Cap letter 0 14 Area uner Stanar Curve P (1 < < 3) 0 z How to fin the proportion of the are uner the stanar normal curve below z or say P ( < z )? P ( > 3) Use Stanar Table!!! 1 16 Stanar P( > 0.3) Area above Areas in the upper tail of the stanar normal istribution

4 Stanar Stanar P(0 < < 0.3) Area between 0 an.3.16 Areas in the upper tail of the stanar normal istribution P(< 0.3) Area below.3.66 Areas in the upper tail of the stanar normal istribution Area Area P ( < < 1.00 ).6 P ( -.00 < <.00 ) P ( < < 3.00 ) P ( < <.33 )

5 Stanarize the 1 Stanarize the N (, ) a b N ( 0, 1) a 0 Stanar b 0 One table! a b P ( a < < b) P < < 6 10 Stanarizing For a normal istribution that has a mean an s.. 10, what percentage of the istribution is between an 6.? 10 Stanarizing P( 6.) P(0.1) Obtaining the Probability Probability Table (Portion) Area P(3. ) P(3. ) P(.1 0) Area

6 10 P(.9 7.1) P(.9 7.1) P(.1.1) 1 10 P( > ) P( > ) P( >.30) Area Area P( > ) More on 10 6% 3% Value is the 6 n percentile 33 The work hours per week for resients in Ohio has a normal istribution with hours & 9 hours. Fin the percentage of Ohio resients whose work hours are A. between & 60 hours. P( 60)? B. less than 0 hours. P( 0)? P( 60)? P( 60) P(0 ) % P( 0)? P( 0) P(.44) %

7 What is z given P( < z) ).0?.0 Fining Values.0 0 z.4 Upper Tail Area z.4 Probability Table (Portion) Fining Values : The weight of new born infants is normally istribute with a mean 7 lb an stanar eviation of 1. lb. Fin the 0th percentile. Area to the left of 0th percentile in In the table there is a area value 0.00 corresponing to a z-score of.4. 0th percentile x lb 3 Fining Values : The Boy Mass Inex for a particular population is normally istribute with a mean an stanar eviation of 4. Fin the 0th percentile. Area to the left of 0th percentile in In the table there is a area value 0.00 corresponing to a z-score of.4. Fining Values : The Boy Mass Inex for a particular population is normally istribute with a mean an stanar eviation of 4. Fin the 40th percentile th percentile +.4 x 4.36 Fining Values : The Boy Mass Inex for a particular population is normally istribute with a mean an stanar eviation of 4. Fin the 40th percentile. Area to the left of 40th percentile in In the table there is a area value 0.40 corresponing to a z-score of.. Stanine Score (Stanar Nine) ? % 4% 4% th percentile. x 4 1-7

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