5.2. Sampling distribution of a sample proportion

5.2.1. 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

.

5.2.2. Sampling distribution of . Its mean and standard deviation

Just like the sample mean, , the sample proportion is also a random variable. Hence, it possesses a probability distribution, which is called its sampling distribution.

It can be shown by relying on the definition of the mean that the mean value of - that is, the mean of all possible values of is equal to the population proportion p just as the mean of the sampling distribution.

Definition:

The mean of the sample proportion is denoted by and is equal to the population proportion p. Thus,

.

The mean of all possible values is equal to the population proportion p.

Since p, the sample proportion is an unbiased estimator of the population proportion.

Now we are interested in determining the standard deviation of the values. Just as in the case of sample mean, , the standard deviation of depends on whether the sample size is a small proportion of the population or not.

Definition:

The standard deviation of the sample proportion is denoted by and defined as

where

p – is the population proportion,

, and n – is the sample size.

This formula is valid when , where N – is the population size.

If , then is calculated as follows:

,

where is called the finite population correction factor.

5.2.3. Form of the sampling distribution of

Now that we know the mean and standard deviation of , and we want to consider the form of the sampling distribution of . Applying the central limit theorem as it relates to the random variable, we have the following:

Definition:

According to the central limit theorem, the sampling distribution of is approximately normal for a sufficiently large sample size.

The random variable

is approximately distributed as a standard normal.

This approximation is good if .

Summary

Let be the sample proportion of success in a random sample from a population with proportion of success p.

Then

1. The sampling distribution of has mean p

.

2. The sampling distribution of has a standard deviation

if

if .

3. The value for a value of is

.

Once again, the last approximation is good if .

Example:

The firm makes deliveries of a large number of products to its customers.

It is known that 75% of all the orders it receives from its customers are delivered on time. Let be the proportion of orders in a random sample of 120 that are delivered on time. Find the probability that the value of will be

a) between 0.73 and 0.80;

b) less than 0.72.

Solution:

From the given information,

, ,

where p is the proportion of orders in the population.

The mean of the sample proportion is

The standard deviation of is

.

Let us find .

.

Since , we can infer from the central limit theorem that the sampling distribution of is approximately normal.

Next, the two values of are converted to their respective Z vales by

.

a) For ; .

For ; .

The required probability is (Figure 5.5).

.

Thus, the probability is 0.7011 that between 73% and 80% of orders of the sample of 210 orders will be delivered on time.

b)

.

Thus, the probability that less than 72% of the sample of 210 orders will be delivered on time is 0.1567.

Exercises

1. For a population, N =40 000 and p = 0.65, find the Z value for each of the following for n = 200

a) =0.59; b) =0.72; c) =0.43; d) =0.73

2. 83% of the households of a large city own VCRs. Let be the population of the households who own VCRs in a random sample of 400 households. Find the probability that the value of will be

a) between 0.85 and 0.88

b) more than 0.80

3. A doctor believes that 80% of all patients having a particular disease will be fully recovered within 3 days after receiving a new drug. Assume that a random sample of 230 patients is selected.

a) What is the mean of the sample proportion of patients?

b) What is the variance of the sample proportion?

c) What is the standard error (standard deviation) of the sample proportion?

d) What is the probability that the sample proportion is less than 0.75?

e) What is the probability that the sample proportion is between

0.78 and 0.85?

4. Sixty percent of adults favor some kind of government control on the prices of medicines.

a) Find the probability that the proportion of adults in a random sample of 200 who favor some kind of government control on the prices of medicines is

i) less than 0.55; ii) between 0.57 and 0.68.

b) What is the probability that the proportion of adults in a random sample of 200 who favor some kind of government control is within 0.04 of the population proportion?

c) What is the probability that the sample proportion is greater than the population proportion by 0.06 or more?

5. Stress on the job is a major concern of a large number of people who go into managerial positions. Eighty percent of all managers of companies suffer from stress. Let be the proportion in a sample of 100 managers of companies who suffer from stress.

a) What is the probability that this sample proportion is within 0.08 of the population proportion?

b) What is the probability that this sample proportion is not within 0.08 of the population proportion?

c) What is the probability that this sample proportion is lower than the population proportion by 0.10 or more?

d) What is the probability that this sample proportion is greater than the population proportion by 0.11 or more?

6. A private university has 1250 students. Of these, 357 concerned about the GPA. A random sample of 265 students was taken.

a) What is the standard error (standard deviation) of the sample proportion of students who are concerned about the GPA?

b) What is the probability that the sample proportion is less than 0.35?

c) What is the probability that the sample proportion is between

0.25 and 0.33?

7. A plant has total of 736 employees. Of these, 342 are married. A random sample of 170 employees was taken.

a) What is the mean of the sample proportion of married employees?

b) What is the standard error of the sample proportion of married employees?

c) What is the probability that the sample proportion is greater than 0.37?

d) What is the probability that the sample proportion is between

0.43 and 0.53?

8. Suppose that 78% of all adults like sport.

a) Find the probability that the proportion of adults who like sport in a random sample of 400 is

i) more than 0.81; ii) between 0.75 and 0.82

iii) less than 0.80; iv) between 0.73 and 0.76

b) What is the probability that the proportion of adults in a random sample of 400 who like sport is within 0.05 of the population proportion?

c) What is the probability that the proportion of adults in a random sample of 400 who like sport is lower than the population proportion by 0.04 or more?

Answers

1. a) -1.78; b) 2.08; c) -6.53; d) 2.37; 2. a) 0.1426; b) 0.9429; 3. a) 0.80;

b) 0.000696; c) 0.0264; d) 0.0294; e) 0.7470; 4. a) i) 0.0735; ii) 0.7974;

b) 0.754; c) 0.0418; 5. a) 0.9544; b) 0.0456; c) 0.0062; d) 0.0030;

6. a) 0.0246; b) 0.9956; c) 0.8906; 7. a) 0.4647; b) 0.0336; c) 0.9976;

d) 0.8223; 8. a) i) 0.0735; ii) 0.8949; iii) 0.8289; iv) 0.8202; b) 0.9844;

c) 0.0268.