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Proportion hypothesis testing

Proportion hypothesis testing is applied for making inferences around a proportion, like for election results. The test holds an assumed proportion up against an alternative claim, like a new sample mean.

The procedure for proportion hypothesis testing is similar to the one described in Hypothesis testing: We state the hypotheses and the significance level (α), calculate the test statistic and take the conclusion.

Proportion hypothesis testing key points

• Proportion hypothesis testing is applied to test an assumption for a population proportion
• It tests new findings against the assumed proportion estimate
• It answers to the question if the new findings are significant with which we can reject the null hypothesis

Procedure for proportion hypothesis testing

The procedure is for proportion hypothesis testing is the same as described in hypothesis testing:

• State null hypothesis
• Sate alternative hypothesis
• State significance level (α)
• Calculate test statistic
• Conclude

I will use the following example to run through the procedure:

A larger organization is due to elect a new board member. It is predicted that candidate C will get 35% of the votes, but running a sample survey of 100 voters, Candidate C becomes 42% of the predicted votes.

Is this new finding of the 42% significant and can the proportion mean therefore be expected to be larger than the assumed 0.35?

Let’s test it proportion hypothesis testing:

Step 1 & 2: State hypotheses

Our null hypothesis states that there is no change. In our voting example, say we wish to test if the proportion mean is greater than the assumed 0.35. The null hypothesis will then state that the proportion is maximum 0.35 despite the new finding of 0.42.

The alternative hypothesis states that the mean proportion is greater than 0.35 as we have had a sample proportion of 0.42. So, the two hypotheses are:

Step 3: State significance level (α)

Second step is to set the significance level (α): We will set it to 0.05, which means that there is a 5% risk that we will reject a correct null hypothesis:

Step 4: Calculation of test statistic

For proportions we can calculate the standard deviation (σ) and we therefore apply the standard normal table. Also, we would usually work with larger sample sizes to work with proportions in the first place.

The z-score is calculated by subtracting the assumed proportion from the sample estimate divided by the standard error of the assumed proportion (the one under the null hypothesis):

Step 5: Conclusion

Our z-score, or our sample statistics, is 1.4676 which is less this than the critical z if 1.96, so we fail to reject the null hypothesis. The test does not give evidence to support the alternative hypothesis and we therefore do not have ground to say that the population proportion is larger than 0.35.

We get a p-value of 0.0711 which is greater than our 0.05 alpha which also leads to the conclusion of failing to reject H0. The p-value means that there is approximately 7.11% chance that we would get at least as extreme a test statistic as the one we had at 0.42 assuming that the true proportion is 0.35.

Proportion hypothesis testing in Excel

In Excel we must calculate the z-score. The critical z can be calculated with the =NORM.S.INV function and the p-value with the =NORM.S.DIST function:

Learning statistics

Some of my preferred material on proportion hypothesis testing

Carsten Grube

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