6.6. Confidence intervals for population proportion: Large samples

The reason leading to estimation of a mean also applies to the problem of estimation of a population proportion.

Suppose that n elements are randomly selected from the large population. And let n consists of X elements possessing some characteristic.

Common sense suggests the sample proportion

as an estimator of p .

When the sample size n is only small fraction of the population size, the sample count X has the binomial distribution with mean and standard deviation .

When n is large, the binomial variable X is well approximated by a normal distribution with mean and standard deviation .

That is

is approximately standard normal.

If we divide both numerator and denominator by n we will get a statement about proportions:

As we see the denominator of Z contains p and q. If sample size is large that , then a good approximation is obtained if p replaces the point estimator in the denominator.

Hence, for large sample size, the distribution of

is approximately standard normal.

We can use this result to obtain confidence interval for the population proportion. (Fig.6.12)

Using Fig.6.12 we obtain

Definition:

If sample of n observations selected from the population is large enough that

, then a confidence interval for the population proportion is given by

where is the sample proportion and is the number for which

.

Example:

From a country labor force a random sample of 800 persons was selected and 75 people were found unemployed out of random sample of 800 persons. Compute 90 % confidence interval for the rate of unemployment in the country.

Solution:

The confidence interval for the population proportion is obtained from

The observed and .

Since , we say that sample size is large and a normal approximation to the sample proportion is justified. Then

and

After substituting, we obtain

Therefore, a 90 % confidence interval for the rate of unemployment in the country is ( 0.0768; 0.1107), or (7.68 %; 11.07 %).

Because our procedure will produce true statements 90 % of the time, we can be 90 % confident that the rate of unemployment is between

7.68 and 11.07.

Exercises

1. Check if the sample size is large enough to use the normal distribution to make a confidence interval for p for each of the following cases

a) n = 60 and

b) n = 180 and

c) n = 200 and

d) n = 65 and

2. A sample of 500 observations selected from a population gave a sample proportion equal to 0.72.

a) make a 90 % confidence interval for p .

b) construct a 95 % confidence interval for p .

c) make a 99 % confidence interval for p .

Interpret your results.

3. A sample selected from a population gave a sample proportion equal

to 0.73

a) make a 98 % confidence interval for p assuming n = 90

b) construct a 98 % confidence interval for p assuming n = 500

c) construct a 98 % confidence interval for p assuming n = 100

Interpret your results.

4. A sample of 87 university students revealed that 53 carried their books and notes in a backpack. Obtain a 95 % confidence interval for the population of students who use backpacks.

5. The Beverage Company has been experiencing problems with the automatic machine that places labels on bottles. A sample of 300 bottles resulted in 27 bottles with improperly applied labels. Using these data, develop a 90 % confidence interval for the population proportion of bottles with improperly applied labels.

6. If 65 persons in a random sample of 180 required lawyer services, then find and interpret 96 % confidence interval for proportion of persons in the population who required a lawyer services.

7. Let sample proportion . How large a sample should be taken to be 95 % sure that the error of estimation does not exceed 0.02 when estimating a proportion?

8. A sample of 20 managers was taken and they were asked whether or not they usually take work home. The responses are given below:

Yes Yes No No No Yes No No

No No Yes Yes No Yes Yes No

No No No Yes

Make a 99 % confidence interval for the percentage of all managers who take work home.

Answers

1. a) Yes, sample size is large; b) No, sample size is not large; c) Yes, sample size is large; d) No, sample size is not large; 2. a) ( 0.687; 0.753);

b) (0.681; 0.759); c) ( 0.668; 0.772); 3. a) (0.621; 0.839); b) (0.684; 0.776); c) (0.627; 0.833); 4. (0.506; 0.712); 5. (0.063; 0.117); 6. (0.286; 0.434);

7. n = 2017; 8. (0.117; 0.683).