Using everything we’ve learned so far about the central limit theorem, the z-score, and hypothesis testing, we can now also perform proportion testing!

There are just a few new concepts to add into the mix:

#### The preliminary terrors – notation & terminology

p = the proportion of items that falls into H_{0}

q = the proportion of items that *don’t* fall into H_{0}

p̂ = the sample proportion

μ = μ_{p̂} = p

σ = √ pq

σ_{p̂} = σ /√ n

Z-score = p̂ – μ_{p̂} / σ_{p̂}

#### Before we begin

You can only do this test successfully IF:

np̂ > 10 *and *nq̂ > 10

If this test fails you will first need to take more samples until both conditions hold true…

And a final note… this test can be used to evaluate against an *actual* population and its parameters OR a hypothesized population and its parameters.

#### A simple example

Your company has been having some problems in manufacturing plastic bags – they keep splitting. The equipment on the production line has been re-calibrated, and it’s expected that this will resolve the issue. You’ve been asked to validate (to a 95% confidence level) that the number of split bags has been reduced to below 4%. The quality control department took a random sample of 400 bags and 13 were found to be split.

In other words we cannot reject the null hypothesis that defects are >= 4%, because the probability (P = 21.78%) is considerably greater the desired confidence level (α = 5%).