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Mean, Variance, and Standard Deviationfor the binomial distribution

For a binomial distribution,

qpn

qpn

pn

2

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ExampleRandomly guessing on a 100 question

multiple-choice test, where each question has 4 possible answers,

3.443

41100

8.1843

41100

2541100

2

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ExampleUsing our range rule of thumb,

6.333.42252

4.163.42252

It would be unusual to get less than 17 questions correct, or more than 33 question correct if you were randomly guessing on the test.

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Another ExampleYou presume that people are evenly split

(50/50) on their preference between two candidates. You poll 80 people, and find 45 prefer candidate A. Is this unexpected?

 We could find P(45 or more).  Alternatively, using our presumed 50/50

preference, find mean=40, stddev=4.5. This suggests that outcomes of 31 to 49 would be usual occurrences. Since 45 falls in that range, we shouldn’t consider it unusual.

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Another ExampleNow suppose you poll 800 people, and 450

prefer candidate A. Is this unexpected? Using our presumed 50/50 preference, mean

= 400, stddev = 14.1. Again using the range rule of thumb, this suggests that outcomes of 372 to 428 would be usual.

Since the outcome of 450 is unusual, we

should question whether our presumption of a 50/50 split is accurate.

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Homework4.4: 1, 5, 7, 11, 13

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