Sampling Distribution of p

Site: Saylor Academy
Course: MA121: Introduction to Statistics
Book: Sampling Distribution of p
Printed by: Guest user
Date: Monday, February 26, 2024, 6:34 AM

Description

Here, we introduce the mean and standard deviation of the sampling distribution of p and the relationship between the sampling distribution of p and the normal distribution.

Sampling Distribution of p

Learning Objectives

  1. Compute the mean and standard deviation of the sampling distribution of p
  2. State the relationship between the sampling distribution of p and the normal distribution

Assume that in an election race between Candidate A and Candidate B, 0.60 of the voters prefer Candidate A. If a random sample of 10 voters were polled, it is unlikely that exactly 60 \% of them (6) would prefer Candidate A. By chance the proportion in the sample preferring Candidate A could easily be a little lower than 0.60 or a little higher than 0.60. The sampling distribution of p is the distribution that would result if you repeatedly sampled 10 voters and determined the proportion (p) that favored Candidate A.

The sampling distribution of p is a special case of the sampling distribution of the mean. Table 1 shows a hypothetical random sample of 10 voters. Those who prefer Candidate A are given scores of 1 and those who prefer Candidate B are given scores of 0. Note that seven of the voters prefer candidate A so the sample proportion (\mathrm{p}) is

p=7 / 10=0.70

As you can see, p is the mean of the 10 preference scores.

Table 1. Sample of voters.

Voter Preference
1 1
2 0
3 1
4 1
5 1
6 0
7 1
8 0
9 1
10 1

The distribution of \mathrm{p} is closely related to the binomial distribution. The binomial distribution is the distribution of the total number of successes (favoring) Candidate A, for example) whereas the distribution of p is the distribution of the mean number of successes. The mean, of course, is the total divided by the sample size, N. Therefore, the sampling distribution of p and the binomial distribution differ in that p is the mean of the scores (0.70) and the binomial distribution is dealing with the total number of successes (7).

The binomial distribution has a mean of:

\mu=N \Pi

Dividing by N to adjust for the fact that the sampling distribution of p is dealing with means instead of totals, we find that the mean of the sampling distribution of p is:

\mu_{\mathrm{p}}=\Pi

The standard deviation of the binomial distribution is:

\sqrt{N \pi(1-\pi)}

Dividing by N because p is a mean not a total, we find the standard error of p:

\sigma_{p}=\frac{\sqrt{N \pi(1-\pi)}}{N}=\sqrt{\frac{\pi(1-\pi)}{N}}

Returning to the voter example, \Pi=0.60 and N=10. (Don't confuse \Pi=0.60, the population proportion and p=0.70, the sample proportion.) Therefore, the mean of the sampling distribution of \mathrm{p} is 0.60. The standard error is

\sigma_{p}=\sqrt{\frac{0.60(1-.60)}{10}}=0.155

The sampling distribution of \mathrm{p} is a discrete rather than a continuous distribution. For example, with an N of 10, it is possible to have a p of 0.50 or a p of 0.60 but not a p of 0.55.

The sampling distribution of p is approximately normally distributed if N is fairly large and \pi is not close to 0 or 1. A rule of thumb is that the approximation is good if both \mathrm{N} \pi and \mathrm{N}(1-\pi) are greater than 10. The sampling distribution for the voter example is shown in Figure 1. Note that even though N(1-\pi) is only 4, the approximation is quite good.


Figure 1. The sampling distribution of p. Vertical bars are the probabilities; the smooth curve is the normal approximation.


Source: David M. Lane, https://onlinestatbook.com/2/sampling_distributions/samp_dist_p.html
Public Domain Mark This work is in the Public Domain.

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Questions

Question 1 out of 3.
The binomial distribution is the distribution of the total number of successes whereas the distribution of p is:

the distribution of the mean number of successes

the distribution of the total number of failures

the distribution of the ratio of successes to failures

a distribution with a mean of .5


Question 2 out of 3.
Out of 300 students in the school, 225 passed an exam. What would be the mean of the sampling distribution of the proportion of students who passed the exam in the school?


Question 3 out of 3.
Out of 300 students in the school, 225 passed an exam. You take a sample of 10 of these students. What is the standard error of p?

Answers


  1. The distribution of the mean number of successes
    The sampling distribution of p is the distribution of the mean number of successes. It has a mean equal to the population proportion.

  2.  .75
    The mean of the sampling distribution of p is equal to the population proportion. It is 225/300 = .75.

  3.   .137
    The standard error of p = sqrt[p(1-p)/N] = sqrt[(.75)(.25)/10] = .137