Proportion Distribution

Definition 1: If x is a random variable with binomial distribution B(n, p) then the random variable y = x/n is said to have the proportion distribution.

Property 1: Where y has a proportion distribution as defined above

image536

Proof: By Property 1b of Expectation and Property 1a of Binomial Distribution

image537

By Property 3b of Expectation

image538

Theorem 1: Provided n is large enough – generally when  np ≥ 5 and (1 – p) ≥ 5 – then N(μy, σy) is a good approximation for the proportion distribution for y with

image541

Proof: This is a consequence of Theorem 1 of the Binomial Distribution.

One Sample Hypothesis Testing

From the theorem, we know that when sufficiently large samples of size n are taken, the distribution of sample proportions is approximately normal, distributed around the true population proportion mean π, with standard deviation (i.e. the standard error)

image542

We can use this fact to do hypothesis testing as was done for the normal distribution. In addition when a two-tailed test is performed a confidence interval can be calculated where

image543 image544

where we use the sample mean p as an estimate for the population mean when calculating the standard error. This introduces additional error, which is acceptable for large values of n.

Example 1: A company believes that 50% of their customers are women. A sample of 600 customers is chosen and 325 of them are women. Is this significantly different from their belief?

H0: π = 0.5; i.e. any difference in the number of men and women is due to chance
H1: π ≠ 0.5

Method 1: Using the binomial distribution, we reject the null hypothesis since:

                BINOMDIST(325, 600, .5, TRUE) =  0.981376 > 0.975 = 1 – α/2 (2-tailed)

Method 2: By Theorem 1 we can also use the normal distribution

The observed mean is 325/600 = 0.541667. Based on the null hypothesis, we can assume that the mean p = .5 and the standard error

image545

         NORMDIST(.541667, .5, .020412, TRUE) = 0.979387 > 0.975 = 1 – α/2 (2-tailed)

And so we reach the same conclusion, namely to reject the null hypothesis.

Example 2: A survey of 1,100 voters showed that 53% are in favor of the new tax reform. Can we conclude that the majority of voters (from the population) are in favor?

H0: π < 0.5

Since people are not surveyed twice, we essentially have a hypergeometric distribution instead of a binomial distribution; i.e. we are selecting without replacement. But for large n the hypergeometric distribution is approximately binomial (i.e. it is not so likely that you will select the same person twice).

Since p = .53 and n = 1100, np > 5 and (1 – p) > 5, and so we can approximate the distribution as a normal distribution. We will also use the sample p as an estimate of π in calculating the standard error.

image5024

NORMDIST(.53, .5, 0.01505, TRUE) = .976889 > .95, and so we can reject the null hypothesis and conclude with 95% confidence that the population will vote in favor of the tax reform.

We determine the 95% confidence interval as follows:

zcrit = NORMSINV(1 – α/2) = NORMSINV(0.975) = 1.96

and so the 95% confidence interval is

image550

Thus we conclude with 95% confidence that between 50.1% and 55.9% of the population will be in favor. If however we are looking for a 99% confidence interval then

 zcrit = NORMSINV(1 – α/2) = NORMSINV(0.995) = 2.58

And so the 99% confidence interval is

image551

This means that with 99% confidence, between 49.1% and 56.9% of the population will be in favor. Since 50.0% is in this interval this time we cannot conclude (with 99% confidence) that the population will vote in favor of the tax reform.

Example 3: In conducting a survey of potential voters, how big does the sample need to be so that with 95% confidence the actual result (i.e. the population mean) will be within 2.5% of the sample mean? (i.e. how big a sample is necessary to have a 2.5% margin of error?)

This time we are looking for the value of n such that

zcrit · s.e. = 2.5%

As we saw in the previous example for 95% confidence zcrit = 1.96. The question is for any value of n when is s.e. the maximum? For any n, s.e. = \sqrt{\frac{p(1-p)}{n}} is maximum when p(1–p) is maximum. It is easy to see that this occurs when p = .5. Thus the maximum s.e. = \sqrt{\frac{.5(1-.5)}{n}} = \frac{.5}{\sqrt{n}}. It now follows that

image558

Solving for n yields n = 1536.584. Thus a sample of size 1,537 is sufficient. Using a similar calculation, to achieve 99% confidence requires a sample of size of 2,654.

Two Sample Hypothesis Testing

Theorem 2: Let x1 and x2 be random variables with proportional distributions with mean π1 and π2 respectively. Let p1 be the proportion of successes in n1 trials of the first distribution and let p2 be the proportion of successes in n2 trials of the second distribution. When the number of trials n1 and n2 are sufficiently large, usually when ni πi ≥ 5 and n(1 –πi) ≥ 5, the difference between the sample proportions p1 – p2 will be approximately normal with mean π1 – π2 and standard deviation

image567

Proof: Based on Theorem 2 of the Binomial Distribution, xi has approximately the distribution

image568

Since x1 and x2 are independently distributed, by the linear transformation property of the normal distribution, x1 – x2 has distribution

image570

Example 4: A company which manufactures long-lasting light bulbs sells halogen and compact florescent bulbs. They ran an experiment in which they ran 100 halogen and 100 florescent bulbs continuously for 250 days. After 250 days they found that half of the halogen bulbs were still working while 60% of the florescent bulbs were still operating. Is there a significant difference between the two types of bulbs?

Let x1 = the percentage of halogen bulbs that are functional after 250 days and x2 = the percentage of florescent bulbs that are functional after 250 days. The presumption is that the distributions for each of these are proportional. We now test the following null hypothesis:

H0: π1 = π2

Assuming the null hypothesis is true, by Theorem 2, x1 – x2 will be approximately normal with mean π1 – π2 = 0 and standard deviation

image572

where the common value of the mean is denoted π and both samples are of size n. Since the value for π is unknown, we estimate its value from the sample, namely, 50 + 60 = 110 successes out of 200, i.e. π ≈ 0.55, Thus, the mean of x1 – x2 is 0 (based on the null hypothesis) and the standard deviation is approximately \sqrt{\frac{2(.55)(.45)}{200}}. The observed value of x1 – x2 is .60 – .50 =.10, and so we have (two-tail test):

         p-value = NORMDIST(.1, 0, .0497, TRUE) = .978 > .975 = 1 – α/2

Thus, we reject the null hypothesis, and conclude there is a significant difference between the two types of bulbs.

We reach the same conclusion since critical value of x1 – x2 = NORMINV(.975,0,.0497) = .0975 < .1 = observed value of x1 – x2.

6 Responses to Proportion Distribution

  1. Guero says:

    Sir,
    It is fair to say, formally, that each of the 600 people asked : X1, X2,..,X600 ,is a proportionally-distributed random variable with mean p and variance p(1-p)/n so that
    by the CLT:

    X= (X1+X2+…..+X600)/600

    Is normally distributed with mean equal to the pop. mean and variance equal to the pop. variance?
    Thanks.

  2. Jonathan Bechtel says:

    Hi Charles,

    I have a question about Example 2.

    To find the confidence interval at 95% I used the Excel equation =CONFIDENCE(0.025, 0.01505, 1100) and got the value 0.00102.

    I’m not sure why this formula is incorrect and doesn’t return the same value as your calculation since they should both be equivalent and I’m not sure how I set up the CONFIDENCE equation incorrectly.

    • Charles says:

      Jonathan,
      I see two problems with the formula =CONFIDENCE(0.025, 0.01505, 1100)
      1. You need to use .05 instead of .05/2 = .025
      2. This formula uses the standard deviation and not the standard error as the second argument. The standard error is then calculated from the standard deviation and the sample size (the third argument).
      Charles

  3. Colin says:

    Sir
    I think you should change ” tcrit ” to “zcrit”

    • Charles says:

      Colin,
      Another great catch. Yes tcrit should be zcrit. I have now made this change on the webpage. Thanks for your help.
      Charles

Leave a Reply

Your email address will not be published. Required fields are marked *