Estimation of population proportions

This form of estimation applies at the times while information cannot be described as a mean or as a measure but only as a percentage or fraction ,The sampling theory stipulates that if repeated large random samples are consider from a population and the sample proportion "p' will be normally distributed along with mean equal to the population proportion and standard error equal to

S_p = √{(pq)/n}  = Standard error for sampling of population proportions

Whereas n is the sample size and q = 1 - p.

The procedure for estimation a proportion is similar to that for estimating a mean; we simply have a different formula for calculating standard.

Illustration 1

In a sample of 800 candidates, 560 were male. Estimate the population proportion at 95 percent confidence level.

Solution

Now

Sample proportion (P) = 560/800  = 0.70

            q = 1 - p = 1 - 0.70  = 0.30

            n = 800

              √{(pq)/n} = √{(0.70)(0.30)/800} 

S_p = 0.016

Population proportion

= P ± 1.96 Sp whereas 1.96 = Z.

= 0.70 ± 1.96 (0.016)

= 0.70 ± 0.03

= 0.67 to 0.73

= between 67 percent to 73percent

Illustration 2

A sample of 600 accounts was taken to test the accuracy of posting and balancing of accounts whereas in 45 mistakes were found. Determine the population proportion. Employ 99 percent level of confidence

Solution

Now

n = 600;  p = 45/600  =  0.075

            q = 1 - 0.075 = 0.925

√{(pq)/n} = √{(0.075)(0.925)/600} 

= 0.011

Population proportion

= P ± 2.58 (S_p)

= 0.075 ± 2.58 (0.011)

= 0.075 ± 0.028

= 0.047 to 0.10

= between 4.7 percent to 10 percent