Presentation is loading. Please wait.

Presentation is loading. Please wait.

Bernoulli Trials Two Possible Outcomes –Success, with probability p –Failure, with probability q = 1  p Trials are independent.

Similar presentations


Presentation on theme: "Bernoulli Trials Two Possible Outcomes –Success, with probability p –Failure, with probability q = 1  p Trials are independent."— Presentation transcript:

1 Bernoulli Trials Two Possible Outcomes –Success, with probability p –Failure, with probability q = 1  p Trials are independent.

2 Binomial Distribution For n Bernoulli trials, the number of successes X is a binomial random variable. The probability of k successes is given by the binomial probability formula: As k varies with fixed n and p, the binomial probabilities define a binomial probability distribution over {0, 1, 2, …, n}.

3 Sampling Distribution of the Count in a SRS When the population is much larger than the sample, the count of X successes in a SRS of size n has approximately the Binomial(n,p) distribution (given that the true proportion of successes in the population is p). As a rule of thumb, we use the binomial sampling distribution for counts when the population is at least 10 times as large as the sample.

4 Mean and Standard Deviation of a Binomial RV X (i.e., of a sample count) Mean and Standard Deviation of a sample proportion,

5 Law of Large Numbers Informal: If n is large, the proportion of successes in n Bernoulli trials will be very close to p. Formal: For Bernoulli trials with n and p, as n  , for all  > 0, where is the sample proportion.

6

7

8 Normal Approximation Draw a SRS of size n from a large population having proportion p of successes. Let X be the count of successes in the sample and = X/n the sample proportion. When n is large, the sampling distributions of the two statistics are approximately normal: As a rule of thumb, we use the approximation for values of n and p such that np  10 and n(1  p)  10.

9 Example 1 – Normal Approximation of Counts Suppose you flip a balanced coin 1000 times. What is the probability of getting between 480 and 532 heads?

10

11

12

13

14 Example 2 – Normal Approximation of Proportions A corporation receives 100 applications for a position from recent college graduates in business. Assuming that these applications constitute a random sample of graduates in business, what is the probability that between 25% and 35% of the applicants are women if 30% of all recent college graduates in business are women?


Download ppt "Bernoulli Trials Two Possible Outcomes –Success, with probability p –Failure, with probability q = 1  p Trials are independent."

Similar presentations


Ads by Google