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Normal distribution and intro to continuous probability density functions... Percent Histogram IQ (Intervals of size 20) 40 Percent 30 20 10 0 55 75 95 IQ 115 135 Histogram (Area of rectangle = probability) IQ (Intervals of size 20) Density 0.02 0.01 0.00 55 75 95 IQ 115 135 Decrease interval size... IQ (Intervals of size 10) Density 0.02 0.01 0.00 55 65 75 85 95 IQ 105 115 125 135 Decrease interval size more…. IQ (Intervals of size 5) 0.03 Density 0.02 0.01 0.00 50 60 70 80 90 100 IQ 110 120 130 140 Continuous probability density functions • The curve describes probability of getting any range of values, say P(X > 120), P(X<100), P(110 < X < 120) • Area under the curve = probability • Area under whole curve = 1 • Probability of getting specific number is 0, e.g. P(X=120) = 0 Special kind of continuous p.d.f 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 . • Spread is determined by . • 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 250 students 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 = Area under curve • Calculus?! You’re kidding, right? • But somebody did all the hard work for us! • We just need a table of probabilities for every possible normal distribution. • But there are an infinite number of normal distributions (one for each and )!! • Solution is to “standardize.” Standardizing • Take value X and subtract its mean from it, and then divide by its standard deviation . Call the resulting value Z. • That is, Z = (X- )/ • Z is called the standard normal. Its mean is 0 and standard deviation is 1. • Then, use probability table for Z. Using Z Table Standard Normal Curve 0.4 Density 0.3 0.2 Tail probability P(Z > z) 0.1 0.0 -4 -3 -2 -1 0 Z 1 2 3 4 Reading Z Table p. 484, Appendix A • Carry out Z calculations to two decimal places, that is X.XX • Find the first two digits (X.XX) of Z in column headed by z. • Find the third digit of Z (X.XX) in first row. • P(Z > z) = probability found at the intersection of the column and row. 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 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 Remember! • Calculated probabilities are accurate only if the assumptions made are indeed correct! • When doing the above calculations, you are assuming that the data are “normally distributed.” • Always check this assumption! (We’ll learn how to next class.)