Hypothesis Testing for a Proportion

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  • 0:03 Background on Proportions
  • 1:22 Conditions for…
  • 3:19 Hypothesis Testing Procedure
  • 5:13 Lesson Summary
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Lesson Transcript
Instructor: Lucinda Stanley

Lucinda has taught business and information technology and has a PhD in Education.

Data sets can be mutually exclusive. What that means is that the population will either be or answer one thing or another. In this lesson, we'll explore how hypothesis testing is applied in that situation.

Background on Proportions

Can you tell by asking 10 of your friends if they prefer chicken or fish as a wedding dinner entrée what the remaining 300 guests will prefer? Well, maybe; what you're trying to do is to determine what a larger population will do based on the results from a smaller population. In statistics, we call that testing for a proportion.

First of all, we need to know what a proportion is. A proportion in statistics is a look at a portion or a part in relation to the whole. It's a comparative relationship between things, meaning if we know what part of a population makes as a choice, we can extrapolate from that to the entire population. In other words, the choices a smaller sample made can be used to determine what the larger population might do.

We can use hypothesis testing for a proportion to perform a statistical analysis to help us answer questions such as:

  • Do more than 80% of married couples have children?
  • Do more than 60% of American men have beards?
  • Do more than 75% of Americans prefer chicken to fish?

In each of these examples, the choices or possibilities are mutually exclusive, meaning either one or the other is true, not both. A married couple either has children or not; Americans prefer chicken or fish; or a man either has a beard or not.

Conditions for Hypothesis Testing

Let's use the chicken or fish preference as an example. The first thing we have to do is make sure the conditions are met to perform a hypothesis test for a proportion. Conditions are things that have to be true in order for the test to be valid or useful. There are four conditions:

1. In order to perform a hypothesis test for a proportion, the sampling method has to be random, meaning the part of the overall population that you are using for your sample has to be chosen randomly: everyone in the population has an equal opportunity to be chosen to participate.

2. Each sample must result in just two outcomes; the outcomes have to be mutually exclusive. Either they prefer chicken or they prefer fish.

3. The sample includes at least 10 successes and 10 failures. This means that at least 10 of each outcome have to be present, so at least 10 people in the sample have to have chosen chicken and at least 10 people in the sample have to have chosen fish.

4. The population size is at least 20 times as big as the sample size. This is a tough one. If the sample size is 100, that means the population size has to be at least 20 times that, or at least 2,000.

If all of these conditions are met, then you are ready to state both a null and alternative hypothesis. So in our chicken versus fish example, the null hypothesis would look like this:

H0 : p >= .75 (the p value is greater than or equal to 75%)

Our null hypothesis essentially states that we think 75% or more of the population will prefer chicken over fish.

And the alternative hypothesis would look like this:

Ha : p < .75 (the p value is less than 75%)

So the alternative hypothesis says that less than 75% of people prefer chicken over fish. p refers to the probability of an outcome. Now that we have our hypotheses, we can begin the process to see which of them is correct.


Now that we have a null and an alternate hypothesis, we can go through the steps to solve our problem: Do more than 75% of Americans prefer chicken over fish?

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