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lecture8 - Binomial Distribution(nish Ch 6 Sampling...

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Binomial Distribution ( nish Ch. 6) Sampling Distributions (Ch. 7) Distribution of sample means & probability Standard error Binomial variables are those that have two possible values or categories Examples Coin Toss: heads/tails Gender: Male/Female Parity: Odd/Even Accuracy: Correct/Incorrect Pregnancy: You either are pregnant or you are not pregnant Two categories: A and B Probability of A= (#Possible As)/(Possible Outcomes)=p(A)= p Probability of B= (#Possible Bs)/(Possible Outcomes)=p(B)= q p+q=1 n = number of observations X = number of times A occurs among the n individuals or observations (0 to n) The binomial shows the probability associated with each value of x

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What are A & B? p & q? n? What does X refer to? Heads and tails p=0.5; q=0.5 n = 2 X = number of heads in the 2 tosses How many heads? 0 1.25 2.50 3.75 5.00 0 1 2 Frequency Number of heads
As the number of tosses increases, the distribution approximates a normal distribution. As a general rule, the normal approximation is valid when pn and qn > 10 If the binomial approximates the normal then we can use the normal distribution to determine probabilities for a binomial variable. called the normal approximation But how do we do this? Convert to z Need " and ± Mean: ± = pn Standard Deviation: ± = z = X ± ± ² z = X ± ± ² = X ± pn npq

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Applying the normal distribution is only an approximation The normal distribution is continuous, but the binomial is, by de nition, discrete. e.g., in 2 coin tosses, it would be impossible to get 1.5 heads.
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