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Normal distribution An example from class Histogram for Female Ability to Match Stick Angle 8 7 5 4 3 2 1 19 17 15 13 11 9 7 5 3 0 1 Frequency 6 Classes of number of sticks correctly matched Example Of a Normal Variable HEIGHTS OF MOTHERS CLASS LIMITS(in.) FREQUENCY 52-53 53-54 54-55 55-56 56-57 57-58 58-59 59-60 60-61 61-62 62-63 63-64 64-65 65-66 66-67 67-68 68-69 69-70 70-71 0.5 1.5 1 2 6.5 18 34.5 79.5 135.5 163 183 163 114.5 78.5 41 16 7.5 4.5 2 1052 TOTAL Histogram Of Heights Of Mothers (in.) Frequency 200 150 100 50 0 Height Normal distribution Bell-shaped curve 0.08 Mean = 70 SD = 5 0.07 Density 0.06 0.05 0.04 Mean = 70 SD = 10 0.03 0.02 0.01 0.00 40 50 60 70 Grades 80 90 100 Characteristics of normal distribution • Symmetric, bell-shaped curve. • Shape of curve depends on population mean () and standard deviation (). • Center of distribution is mean () and mode and median. • Spread is determined by standard deviation(). • Most values fall around the mean, but some values are smaller and some are larger. Examples of normal random variables • • • • • • • • testosterone level of male students head circumference of adult females length of middle finger of Stat/Soc students Height Weight IQ scores Body temperature Repeated measurement of same quantity Probability between 65 and 70? 0.08 0.07 Density 0.06 0.05 P(65 < X < 70) 0.04 0.03 0.02 0.01 0.00 55 60 65 70 Grades 75 80 85 Probability above 75? Probability student scores higher than 75? 0.08 0.07 Density 0.06 0.05 P(X > 75) 0.04 0.03 0.02 0.01 0.00 55 60 65 70 Grades 75 80 85 Probability below 65? 0.08 0.07 Density 0.06 0.05 0.04 0.03 0.02 P(X < 65) 0.01 0.00 55 65 75 Grades 85 Normal Percents The 68-95-99.7 Rule Example: Young Women’s Height • The heights of young women are approximately normal with mean = 64.5 inches and std.dev. = 2.5 inches. Example: Young Women’s Height • • • • % of young women between 62 and 67? % of young women lower than 62 or taller than 67? % between 59.5 and 62? % taller than 68.25? Example: Young Women’s Height • The heights of young women are approximately normal with mean = 64.5 inches and std.dev. = 2.5 inches. Example: Young Women’s Height • The heights of young women are approximately normal with mean = 64.5 inches and std.dev. = 2.5 inches. Example: Young Women’s Height • • • • % of young women between 62 and 67? % of young women lower than 62 or taller than 67? % between 59.5 and 62? % taller than 68.25? Working With the General Normal EXAMPLE: IQ Scores IQ Scores have a normal distribution with a mean of 100 and a standard deviation of 16. What is the 99% percentile of IQ Scores? s.d. = 16 | 100 The Standard Normal Table: Table A • Table A is a table of areas under the standard normal density curve. The table entry for each value z is the area under the curve to the left of z.

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