Sampling Distribution of the Proportion: Estimating Population Proportions, Lecture notes of Technology

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THE “SAMPLING DISTRIBUTION

OF THE PROPORTION”

Instead of measuring

numerical values sometimes

we count, or use the

proportion of a population

that possesses a particular

characteristic.

Researchers study the

proportion of:

• U.S. adults who wear seat belts.
• individuals contracting H1N

virus.

• one parent families.
• people who will vote Democratic.
• people ordering I-pads / new

technology.

THE “SAMPLING DISTRIBUTION

OF THE PROPORTION”

The sample proportion is the ratio of the number of occurrences in the sample to the sample size. The sample proportion, denoted (read p-hat), is expressed:

x = number of occurrences in the sample n = sample size

n

x p ˆ 

p ˆ

THE SAMPLING DISTRIBUTION

OF THE PROPORTION

each group is the ratio in the bag of candy population

pn

p

p

p

p

p

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

5

4

3

2

1



The Sampling Distribution of

the Proportion is…

• a probability distribution
• all the possible values of the sample

proportions and probabilities listed

• all samples are the same size
• all are selected from a population

Ex: Proportions of Blue candy in each

bag of M&M’ s

Notation for the Mean of the

Sampling Distribution of the

Proportion

The mean of the sampling distribution of

the proportion:

read “ mu sub p hat ”

 p ˆ

Theory!

The Mean of the Sampling Distribution

of the Proportion = mean of the

population proportion:

Standard Error of the Proportion:

n

pq SE or  p ˆ 

 p ˆ  P

Sampling Error of the

Proportion

Assuming the sample is random and there

are no non-sample errors.

The sampling error for the proportion is:

sampling error is the difference

sample proportion – population proportion

In symbols:

p ˆ^  P

Example #

According to the National Highway

Traffic Safety Administration, 70%

of all motor vehicle drivers within

use their seat belts. Assume this

proportion is true for the

population of all motor vehicle

drivers who use their seat belts. If

a random sample of size 200 is

selected from this population, then

determine the:

b) Standard error of the

proportion (to 4 decimal places)

ˆ ^  

n

pq  p

c) Shape of the sampling

distribution of the proportion

Normal

ˆ ^  

n

pq  p

 p ˆ  0.

1. 0446

2. 7

3. 0353

4. 53 0. 47 ˆ





 

table

Z (^) p