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THE BINOMIAL DISTRIBUTION THE BINOMIAL DISTRIBUTION • A discrete distribution THE NORMAL DISTRIBUTION • A continuous distribution APPROXIMATION OF THE BINOMIAL BY THE NORMAL DISTRIBUTION Page 1 of 12 Part 3 THE BINOMIAL DISTRIBUTION CONSIDER AN EVENT A WITH A KNOWN PROBABILITY p EXAMPLE • in a clinical trial the probability of adverse events is 25% ONE CONSIDERS n EXPERIMENTS (n patients) EACH TIME EITHER A OCCURS OR IT DOES NOT OCCUR CALL i THE NUMBER OF OCCURRENCES EXAMPLE • Suppose 20 patients were included • Calculate the probability that the event occurs i times Page 2 of 12 Part 3 THE BINOMIAL DISTRIBUTION Probability of i side effects (0.25)i . (0.75)20-i P(i) = where = 20 x 19 x 18 x 17 x ............ x 2 x 1 i.(i-1).(i-2) .1. (20-i).(20-i-1). 2.1 (0.25)i = 0.25x0.25x.....x0.25 [i factors] (0.75)20-i = 0.75x0.75x.....x0.75 [20-i factors] Page 3 of 12 Part 3 THE BINOMIAL DISTRIBUTION EXAMPLE Probability that exactly 6 patients suffered from adverse events (0.25)6 . (0.75)14 P(6) = = 20! . = 38760 6! x 14! (0.25)6 = 0.000244 (0.75)14 = 0.017818 P(6) = 0.1686 Page 4 of 12 Part 3 THE BINOMIAL DISTRIBUTION Other Probabilities P(0) = 0.169 P(1) = 0.169 P(2) = 0.169 P(3) = 0.169 P(4) = 0.169 P(5) = 0.169 P(6) = 0.169 P(7) = 0.169 P(8) = 0.169 P(9) = 0.169 P(10) = 0.169 P(11) = 0.169 P(12) = 0.169 These probabilities depend upon n and p Page 5 of 12 Part 3 THE BINOMIAL DISTRIBUTION PARAMETERS mean value = n.p variance = n.p.(1-p) mean value = n.p.(1-p) Page 6 of 12 Part 3 APPROXIMATION OF A BINOMIAL BY A NORMAL DISTRIBUTION f(x) = where x = random variable (pulse rate) e = 2,72 µ = mean value σ = standard deviation obtained by: decreasing the class interval increasing the sample size Page 7 of 12 Part 3 APPROXIMATION OF A BINOMIAL BY A NORMAL DISTRIBUTION The normal distribution is determined by the mean value: µ the standard deviation: σ Any normal distribution can be simplified into a particular normal distribution by the transformation X-µ Z = --------------σ The new normal distribution has mean value: 0 standard deviation: 1 Page 8 of 12 Part 3 APPROXIMATION OF A BINOMIAL BY A NORMAL DISTRIBUTION This transformation is called a Z-transformation The new distribution is a standardized normal distribution It is denoted as N(0,1) Page 9 of 12 Part 3 APPROXIMATION OF A BINOMIAL BY A NORMAL DISTRIBUTION PROPERTIES OF THE N(0,1) DISTRIBUTON P[-0.67 ≤ Z ≤ 0.67] P[-1 ≤Z≤1 = 50.0 % ] = 68.3 % P[-1.96 ≤ Z ≤ 1.96] = 95.0 % P[-2 ≤Z≤2 ] = 95.4 % P[-3 ≤Z≤3 ] = 99.7 % Page 10 of 12 Part 3 APPROXIMATION OF A BINOMIAL BY A NORMAL DISTRIBUTION APPLICATION OF THE N(0,1) DISTRIBUTON Laboratory results from different laboratories (with different normal ranges) can be made comparable if • the variable has a normal distribution • the normal ranges correspond to 95% tolerance ranges EXAMPLE LAB 1 LAB 2 40-80 40-76 mean 60 58 st. dev. 10 9 test result 50 50 normal range standardized (50-60)/10=-1 Page 11 of 12 (50-58)/9=-0.89 Part 3 APPROXIMATION OF A BINOMIAL BY A NORMAL DISTRIBUTION For LARGE n the shape of a binomial distribution is approximately the same as the shape of a normal distribution CONDITIONS n.p≥ 5 n . (1-p) ≥ 5 Page 12 of 12 Part 3