# Test a proportion against unanimity

How can I test a proportion against an expected value of 1.0?

Made up example (slightly silly):

Dichotomous Yes/No question...

Would you like me to hit you very hard on the head with this hammer?

Expected proportions: No = 1.00, Yes = 0.00

Observed proportions: No = 0.85, Yes = 0.15

How can I test if the observed No proportion of 0.85 is different from the expected No proportion of 1.0?

Can't use a one proportion z test as I will get 0 for my denominator.

Any help gratefully received.

Mike

P.S. The real example from my student's project would have taken too long to describe.

## 1 Answer

You should include the information on how many actual yes's and no's you've collected and test those against each other. If for example, if you expect 50% and you've collected 1-3 it's a different story from seeing 100-300, in the latter case you'd be pretty sure it's not 50% while in the former that could still be a fine guess.

If you want to do hypothesis testing, you should use a https://en.wikipedia.org/wiki/Binomial_test.

It's interesting if you are expecting a proportion of exactly 0 or exactly 1, because those you would be able te reject immediately after seeing one counterexample. Is that really your null hypothesis?

• Hi. Yes, the null hypothesis is really a proportion of 1.0. The real example is a very basic question to health professionals that you would expect all of them to get correct. For example if you asked 100 doctors: Is the fibia in the arm or the leg? You would expect them 100 get this right. It was a kind of screening question and surprisingly quite a few participants got the answer wrong. Commented Jan 29, 2019 at 10:16
• And that the proportion was not 1.0 is a little bit scary, suggesting that a number of these health professionals lack basic knowledge you would expect them to have. Obviously as it was an online questionnaire some may have accidentally have clicked on the wrong answer, so the student and I want to be confident that the proportion is significantly different to 1.0. Thanks for your interest. Commented Jan 29, 2019 at 10:21
• Fibula, I meant fibula. Commented Jan 29, 2019 at 10:23