6.6 normal approx p hat

6.6 NORMAL APPROXIMATION TO BINOMIAL DISTRIBUTION AND TO P DISTRIBUTION

Chapter 6:

Normal Curves and Sampling Distributions

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Normal Approximation to the Binomial Distribution

Under the conditions stated below, the normal distribution can be used to approximate the binomial distribution.

Consider a binomial distribution wheren = number of trialsr = number of successesp = probability of success on a single trialq = 1 – p = probability of failure on a single trial

If np > 5 and nq > 5, then r has a binomial distribution that is approximated by a normal distribution with

and Note: as n increases, the approximation becomes better

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Example 14 – Binomial Distribution Graphs

Notice that as n increases, the normal approximation

to the binomial distribution improves

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Continuity Correction

The normal distribution is for a continuous random variable. The binomial distribution is for a discrete random variable. So, in order to use the normal distribution to approximate the binomial distribution, we need to make a continuity correction.

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How to Make the Continuity Correction

Convert the discrete random variable r (number of successes) to the continuous normal random variable x by doing the following:

1. If r is a left point of an interval, subtract 0.5 to obtain the corresponding normal variable x.

x = r – 0.5

2. If r is a right point of an interval, ass 0.5 to obtain the corresponding normal variable x.

x = r + 0.5

Example:P(6 ≤ r ≤ 10) would be approximated by P(5.5

≤ r ≤ 10.5)

How to Make the Continuity Correction

How to Find Probabilities

Given a binomial distribution wheren = number of trials

r = number of successes

p = probability of success on a single trial

q = 1 – p = probability of failure on a single trial

np > 5

nq > 5

1. Define what you are trying to find2. Make the continuity correction3. Convert to z scores4. Use the standard normal distribution to find

the corresponding probabilities

Not in Textbook!

Note in order to do this, you must find μ and σ.

Example 15 – Normal Approximation

The owner of a new apartment building must install 25 water heaters. From past experience in other apartment buildings, she knows that Quick Hot is a good brand. A Quick Hot heater is guaranteed for 5 years only, but from the owner’s past experience, she knows that the probability it will last 10 years is 0.25.

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a) What is the probability that 8 or more of the 25 water heaters will last at least 10 years? Define success to mean a water heater that lasts at least 10 years.

Example 15 – Normal Approximation

Solution:

n = 25r = binomial random variable corresponding to the number of successesp = 0.25q = 0.75np = 6.25nq = 18.75

We want: P(r≥8)

= P(r≥7.5)

= Normalcdf(.58, E99)= .280957≈ .2810

The probability that 8 or more of the 25 water heaters will last at least 10 years is approximately .2810.

Sampling Distributions for the Proportion p

Givenn = number of binomial trials (fixed constant)

r = number of successes

p = probability of success on each trial

q = 1 – p = probability of failure on each trial If np > 5 and nq > 5, then the random

variable can be approximated by a normal random variable (x) with mean and standard deviation

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The standard error for the distribution is the standard deviation

We do not use a continuity correction for the distribution.

is an unbiased estimator for p, the population proportion of success.

Sampling Distributions for the Proportion p

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Example 16 – Sampling Distribution of p

The annual crime rate in the Capital Hill neighborhood of Denver is 111 victims per 1000 residents. This means that 111 out of 1000 residents have been the victim of at least one crime. These crimes range from relatively minor crimes (stolen hubcaps or purse snatching) to major crimes (murder). The Arms is an apartment building in this neighborhood that has 50 year round residents. Suppose we view each of the n = 50 residents as a binomial trial. The random variable r (which takes on values 0, 1, 2, . . . , 50) represents the number of victims of at least one crime in the next year.

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Example 16 – Sampling Distribution of p

a) What is the population probability p that a resident in the Capital Hill neighborhood will be the victim of a crime next year? What is the probability q that a resident will not be a victim?

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Solution:

Example 16 – Sampling Distribution of p

b) Consider the random variableCan we approximate the distribution

with a normal distribution? Explain.

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Solution:

Since both np and nq are greater than 5, we can approximate the distribution with a normal distribution.

Example 16 – Sampling Distribution of p

c) What are the mean and standard deviation for the distribution?

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Solution:

Assignment

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