8.2 SAMPLING DISTRIBUTIONSDistribution of the Sample Proportion

Obj: Use sample data distribution to approximate probability

SAMPLE DISTRIBUTIONS

Distribution of the Sample Mean

Whether the spread is normal or not, as long as the population is greater than 30,

μx = μ

σx = n

DISTRIBUTION OF THE SAMPLE PROPORTION

The sample proportion estimates the population proportion.

If 16 of a population of 100 have a certain characteristic, the population proportion is 16/100 = 0.16.

The sample proportion p = where x is the number

of individuals and n is the random sample size.

n

x

PROPERTIES OF THE SAMPLING DISTRIBUTION OF P

If n < 0.05N, then…

The shape of the distribution is approximately normal as long as np(1 - p) > 10

The mean of p is μp = p

The standard deviation σp = n

pp )1(

EXAMPLE

Describe the sampling distribution of p. Assume the size of the population is 25000.

n = 500 and p = 0.4

Is n < .05N? Is np(1 – p) > 10?500 < .05(25000)? 500(.4)(.6) > 10?Yes, so the sampling distribution is approximately normalμp = 0.4

σp = 022.0500

)6(.4.

FINDING PROBABILITY

A nationwide study in 2003 indicated that about 60% of college students with cell phones send and receive text messages with their phones. Suppose a simple random sample of n = 1136 college students with cell phones is obtained.

a) Describe the sampling distribution of p.Normal? Mean? Standard Deviation?

b) What is the probability that 665 or fewer college students in the sample send and receive text messages with their cell phones?

c) What is the probability that 725 or more send or receive messages?

PRACTICE

Peanut and tree allergies are considered to be the most serious food allergies. According to the National Institute of Allergy and Infectious Diseases, roughly 1% of Americans are allergic to peanuts or tree nuts. Suppose a random sample of 1500 Americans is obtained.

(There are approximately 295 million Americans.)

Describe the sampling distribution of p.

What is the probability that more than 1.5% are allergic to peanuts or tree nuts?

PRACTICE

According to the National Center for Health Statistics (2004), 22.4% of adults are smokers. Suppose a random sample of 300 adults is obtained.

Describe the sampling distribution of p.

In a random sample of 300, what is the probability that at least 50 are smokers?

Would it be unusual if a random sample of 300 results in 18% or less being smokers?

ASSIGNMENT

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