Normal Approximation to Binomial Distribution

Normal Approximation to Binomial Distribution

Consider the binomial distribution with n trials, and probability of success is p

This distribution is approximately normal if

• np > 5 and nq > 5.In this case it is approximated by a

normal distribution with • Mean = np and Variance = npq

Consider the binomial distribution with n trials, and probability of success is p

This distribution is approximately normal if

• np > 5 and nq > 5.In this case it is approximated by a

normal distribution with • Mean = np and Variance = npq

Binomial Histogram (n=25, p=.95)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

2 4 6 8 10 12 14 16 18 20 22 24

Number of Successes

Probability

This binomial distribution doesn’t look approximatelynormal (it is not bell shaped). Note np = 23.75 > 5 but nq = 1.25 < 5.

Binomial (n=25, p=.7) and Normal Approximation

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 3 5 7 9 11 13 15 17 19 21 23 25

Number of Success

Probability

The normal distribution is a good approximation: np = (25)(.7) = 17.5 > 5, nq = (25)(.3) = 7.5 > 5

Binomial (n=40, p=.8) and Normal Approximation

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 4 7 10 13 16 19 22 25 28 31 34 37 40

Number of Success

Probability

The normal is a good approximation: np = 32> 5 and nq = 8 > 5.

The normal is a good approximation: np = 40> 5 and nq = 60 > 5.

Binomial (n=100, p=.4) and Normal Approximation

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69

Number of Success

Probability

Ex: For the following distribution, estimate P(9 < r < 13) by(i) using the binomial probability formula directly;(ii) using the normal approximation.

Continuity CorrectionContinuity Correction

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