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1.2.3 Sampling Distribution of Proportions ( P )
Sometimes in statistics, it is important to know the proportion of a certain characteristic in a
population. That is, there are numerous problems in business where we want to know the
proportion of items in a population that possess a certain characteristic. For example, a quality
control engineer might want to know what proportions of products of an assembly line are
defective. A labor economist might want to know what proportion of the labor force is
unemployed.
Whereas the mean is computed by averaging a set of values, the sample proportion is computed
by dividing the frequency that a given characteristic occurs in a sample by the number of items
in the sample.
n
X
P
Where (^) P = sample proportions X = number of items in a sample that possess the characteristic n = number of items in the sample
Like other probability distribution, sampling distribution of the proportion is described by two
parameters: the mean of the sample proportions, E ( P ) and the standard deviation of the
proportions, P which is called the standard error of the proportion.
To determine how close the sample proportion is to the population proportion p , we need to
understand the expected value of, the standard deviation of, and the shape or form of the
sampling distribution of p.
Expected Value of p
The expected value of p , the mean of all possible values of p , is equal to the population
proportion p.
E ( p ) = p
Where E ( p ) = the expected value of p
p = the population proportion
Example : A research institute study about managers who participated in management training
program and have a proportion of 60%. The expected value of p for the sampling problem is 60
Standard deviation of p
Just like the standard deviation of mean, the standard deviation of p depends on whether the
population is finite or infinite. The two formulas for computing the standard deviation of p are
as follows:
Example : Let’s take the previous situation again. The population proportion of managers who
participated in the management training program is p = .60 with sample size of 30 and
population of 2500. n / N = 30/2500 = 0.012, we can ignore the finite population correction factor
when we compute the standard error of the proportion. For the simple random sample of 30
managers, is
A research institute study about managers who participated in management training program and
have a proportion of 60%. The expected value of p for the sampling problem is 60 %.
Form of the sampling distribution of p
The sample proportion is p= x/ n. For a simple random sample from a large population, the
value of x is a binomial random variable indicating the number of elements in the sample with
the characteristic of interest. To determine whether the sample size is large enough and sampling
distribution of sample proportion is approach to normal distribution, it must satisfy the following
two conditions: np ≥ 5 and n (1- p )≥ 5. The sampling distribution of p can be approximated by a
normal distribution whenever np ≥ 5 and n (1 - p ) ≥ 5.
Exercises
- The mean undergraduate cost for tuition, fees, room, and board for four-year institutions was $26,489 for a recent academic year. Suppose that standard deviation of $3204 and that of 36 four-year institutions are randomly selected. Find the probability that the sample mean cost for these 36 schools is A. Less than $25, B. Greater than $26, C. Between $24,000 and $26, D. Find the standard error of the sampling distribution
- Time spent using e-mail per session is normally distributed, with mean of 8 minutes and standard deviation of 2 minutes. If you select a random sample of 25 sessions,
A. What is the probability that the sample mean is between 7.8 and 8.2 minutes? B. What is the probability that the sample mean is between 7.5 and 8 minutes? C. If you select a random sample of 100 sessions, what is the probability that the sample mean is between 7.8 and 8.2 minutes?
- Suppose that during any hour in a large department store, the average number of shoppers is 448, with a standard deviation of 21 shoppers. What is the probability of randomly selecting 49 different shopping hours, counting the shoppers, and having the sample mean fall between 441 and 446 shoppers, inclusive?
- The U.S. Census Bureau announced that the median sales price of new houses sold in 2009was $215,600, and the mean sales price was $270,100. Assume that the standard deviation of the prices is $90,000.
A. If you select a random sample of n=100 what is the probability that the sample mean will be less than $300,000? B. If you select a random sample of n= 100 what is the probability that the sample mean will be between $275,000 and $290,000?
- A population has a mean of 200 and a standard deviation of 50. Suppose a simple random sample of size 100 is selected and is used to estimate μ. A. What is the probability that the sample mean will be within +5 of the population mean? B. What is the probability that the sample mean will be within +10 of the population mean?
- A population proportion is .40. A simple random sample of size 200 will be taken and the sample proportion will be used to estimate the population proportion. A. Compute the standard error of the proportion B. What is the probability that the sample proportion will be within+.03 of the population proportion? C. What is the probability that the sample proportion will be within +.05 of thepopulation proportion?
- In a recent survey of full-time female workers ages 22 to 35 years, 46% said that they would rather give up some of their salary for more personal time. Suppose you select a sample of 100 full-time female workers22 to 35 years old. A. What is the probability that in the sample, fewer than 50% would rather give up some of their salary for more personal time? B. What is the probability that in the sample, between 40% and 50% would rather give up some of their salary for more personal time? C. What is the probability that in the sample, more than 40% would rather give up some of their salary for more personal time?
