Population Proportion is the ratio of people who possess a certain characteristic in the given population to the total number of the population.
By a ‘characteristic’ we simply mean any property that the individual in a given population may have.
Some examples of characteristics are eye color, sex, health status (sick vs healthy), etc. For example, in a class of 50 students, we may be interested in the proportion of people who are female.
Formula for Population Proportion:
Population proportion = Number of people with given characteristic/Total population
Estimating the Population Proportion:
If the population size is very large then the population proportion may be difficult to calculate. In such cases, we use the sample proportion as a point estimate for the population proportion.
For example, suppose we want to know the proportion of Americans who are blue-eyed. Checking the eye color of the entire population will be practically impossible.
In such a case it is reasonable to choose a random sample of Americans and find the proportion of blue-eyed Americans in the sample and use that as an estimate for the population proportion.
Confidence interval for Population Proportion:
We can also use the sample proportion to calculate the 95% confidence interval for the population proportion.
The 95% confidence interval gives us an interval within which the true value of the population proportion lies (up to 95% accuracy).
The procedure to calculate the confidence interval for a population proportion is as follows,
- Find out the sample proportion (denoted as small p). This is done by dividing the number of people who possess the given characteristic in our sample by the total sample size.
- Find the proportion of people in our sample who do not possess the given characteristic (denote it as q). The formula is q = 1-p
- Calculate the 95% confidence interval using the formula,
(p-1.96*√(p*q∕n), p+1.96*√(p*q∕n))
Example:
Suppose that in a given city we choose a random sample of 100 people out of which 30 of them are blue-eyed. Calculate the 95% confidence interval for the proportion of blue-eyed people for the entire population of the city.
Solution: Here sample size n=100.
Since 30 out of 100 people have blue eyes, therefore,
Sample proportion p= 30/100 = 0.3
And q = 1-p = 1-0.3 = 0.7
The 95% confidence interval is given as,
= (p-1.96*√(p*q∕n), p+1.96*√(p*q∕n))
= (0.3-1.96*√(0.3*0.7∕100), 0.3+1.96*√(0.3*0.7∕100))
= (0.3-1.96*√0.0021, 0.3+1.96*√0.0021)
= (0.21, 0.39)
So the proportion of blue-eyed people in the city lies between 0.21 and 0.39 (with 95% accuracy).
