IMPORTANT - PLEASE READ
As you saw on the previous slide, the sampling distribution for \(\hat p\) depends on \(n\) and \(p\) . When \(p\) is closer to 0.5, and \(n\) is larger, the sampling distribution looks more Normal.
Here's a summary: Choose an SRS of size \(n\) from a population of size \(N\) with proportion \(p\) of successes. Let \(\hat p\) be the sample proportion of successes. Then, the following is true for the sampling distribution of \(\hat p\) (as long as the conditions are met):
| |
Formula/Attribute |
Condition that must be met |
| Shape: |
approximately Normal |
\(np \ge 10\) and \(n(1-p) \ge 10\) |
| Center: |
\(\mu_\hat p = p\) |
Random sampling (usually an SRS) |
| Spread: |
\(\sigma_ \hat p = \sqrt \frac {p(1-p)}{n}\) |
\(10n \leq N\) (10% condition) |
We will check these 3 conditions EVERY TIME we do a sampling problem about proportions.
