Probability that P > 0.1 of a population from a small sample

Hey guys!

I'm having an issue here with calculating the probably that the proportion of a population is larger 10% using data from a small sample.

For example:
Hypothesis: more than 10% of the population own a smartwatch.
Sample size n = 30
# successes x = 3
p^ = 0.1

How would I calculate the confidence level that the actual value of P is larger than 0.1 for the population?


Omega Contributor
Well, what you can do is slap 95% proportion confidence intervals (CI) on your estimate. Next, you can use the CI values to make a statement such as, we are 95% confident the true proportion of persons with a smartwatch is between lower interval value and upper interval value.

This calculation takes into account your sample size. Also, you do not have to use 95% CI, you could use 99% CI, though per logic and the formula, to be more confident you want to be the wider the interval will likely become.
Hi hlsmith!

Thanks for you answer.

I am aware that I can say that the real P is between the margins of p^ with the chosen confidence level. However, how would I be able to answer the question "What is the probability that the value of P is larger than 0.1"?
Indeed. I want to know something about the P of the population, however, I am interested in the probability that P is larger than 0.1.

Right now, I understand how to get the confidence interval and the certainty that the actual P is within this confidence interval. However, I still do not know what the probability is that the actual P is larger than 0.1.


Omega Contributor
The actual P is what it is, it doesn't have a probability of being bigger than something. Your pursuits deals with samples given the population. So what is the probability a sample value is larger than x given the population value is X.

The true value (population) is say 10 people out of 100, so 10/100, thus the probability of having a watch is .1, there is not uncertainty about that.

I'm sorry, I think I made myself unclear.
With P i am not referring to the value of the population (N). With P i mean the proportion of the population that has a smartwatch.
In my research, the assumption is that N is sufficiently large, that we can consider it infinite.

For example, in my previous example, p^ was 0.1. However, this does not mean that 10% of the population has a smartwatch. The proportion (P) of the population that has a smartwatch is 0.1 +/- a confidence interval with a certain confidence level. However, how do I calculate that confidence level that the proportion (P) of the population that has a smartwatch is larger or equal to 0.1, given the results of the experiment?

I want to thank you again for your help! :)


Omega Contributor
Sample = subset of the population

Population = all possible people of interest

You use statistics with sample data (e.g., confidence intervals, etc.). You just report values for populations, because there is no uncertainty if you have every single person. If you have a population, why would you ever not be confident about the value. There would be absolutely no suspected variability in the population value, especially if you are writing about a proportion.
Would this equation work or do you see any problems with it?

n = sample size
x = number of people in sample that has a smartwatch
P = true proportion of population that has a smartwatch