# Hypothesis testing for one sample proportions

$H_0: p = 0.1$

$H_a: p \neq 0.1$

So here I have a pretty small sample, and I read somewhere that in order to approximate this binomial distribution with the normal distribution, I have to have $n*p > 5$ and $n*(1-p) > 5$. Since $30 * 0.1 = 3 < 5$, does that mean I cannot use the $Z$ statistic here? If so, what can I do to overcome this?

• The Binomial distribution gives you an exact distribution for the number of successes in $n$ trials when these are independent and each trial has success probability $p$. – Christoph Hanck May 28 '15 at 7:38
• @Adrian You don't need the normal approximation. You're doing an exact test and can derive the p-value directly. Use that p-value. Google "Exact test". – SmallChess May 28 '15 at 8:05
• two tailed binomial test – Glen_b May 28 '15 at 13:18

## 1 Answer

You can do an exact calculation using the binomial theorem. If p=.1 and n=30, what is the probability of 5,6,7 ... 30 cancer cells. Add these probabilities together (which will be .175). That's a one-tailed test. Double this probability for two tails.

The individual probabilities can easily be calculated in Excel using the BINOMDIST function.

• Once you get the probability (say it's 0.175). What do I do next? How do I know if I reject or fail to reject the null? – Adrian May 28 '15 at 7:49
• If the conventional alpha level for this research is .05, two tailed, you have 2 (.175) = .35 here, which is substantially higher than .05. So don't reject the null, if these are your actual numbers. – zbicyclist May 28 '15 at 19:52
• I'm a little confused why you're doubling a one tailed probability here. What's the actual two-tailed rejection region? – Glen_b Oct 5 '15 at 23:08
• @Glen_b you are correct that there is no rejection region at the low end with alpha=.05. With p=.1, n=30 getting 0 events has a probability of .042. I was following this logic: a two-tailed test is generally appropriate, logically. If so, you would want to locate half the rejection region at the high end, half at the low. So if you have .175 at the high end for a hypothesis test, that's really a one-tailed test. – zbicyclist Oct 6 '15 at 15:49