Population and sample proportions

The concept of proportion is the same as the concept of relative frequency discussed in Chapter 2 and the concept of probability of success in a binomial distribution. The relative frequency of a category or class gives the proportion of the sample or proportion that belongs to that category or class. Similarly, the probability of success in a binomial problem represents the proportion of the sample or population that possesses a given characteristic.

The population proportion, denoted by p, is obtained by taking the ratio of the number of elements in a population with a specific characteristic to the total number of elements in the population.

The sample proportion, denoted by (read as ” p hat”) gives a similar ratio for a sample.

Definition: The population and sample proportions, denoted by p and , respectively, are calculated as

and

where

total number of elements in the population;

total number of elements in the sample;

number of elements in the population or sample that possesses a specific characteristic.

Example:

Suppose a total of 393 217 families live in a city and 123 017 of them own at least one car. Then,

population size = 393 217

families in the population who own car =123 017.

The proportion of families in this city who own car is

.

Now, suppose that a sample of 560 families is taken from this city and 215 of them have at least one car. Then

sample size =560

families in the sample who own car = 215.

The sample proportion is

.

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