Population Distributions

4 Concepts
Sampling Distributions

3 Concepts
Z-scores

6 Concepts
T-scores

5 Concepts
Assumptions for Sampling Distributions

4 Concepts
Confidence Intervals

8 Concepts
Hypothesis Tests

13 Concepts
**Scenario**: At Crammer Nation University, 60% of students join Greek Life. What's the probability of finding a sample of 30 people with a sample proportion of less than 65%?

Clue | Insight |
---|---|

60% of students join Greek Life is a proportion of students who joined Greek Life. | We're working with proportions here, not means. |

We're being asked to consider a sample of 30 people. | We'll be working with a sampling distributions, not the population distribution. |

Given our sample abides by the assumptions for sample proportions, and therefore the Central Limit Theorem applies, our sampling distribution is normal. | We can use z-scores instead of t-scores. |

What we'll do:

- Compute z-score, given our p-hat value of 0.65
- Find the area under the z-distribution to the left of our z-score (seen in red below)...
- ...which will be equal to the p-value corresponding to our z-score!

**p-hat** is the sample proportion we're testing on.**p** is the population proportion.**q** is the "proportion of failure", a.k.a. the opposite of the population proportion**n **is the sample size.

**Standard Error (SE)** looks a little different with proportions. That's because standard deviation (σ) doesn't occur with proportions. It gets replaced by `√(pq)`

!

We can see Standard Error (SE) still occurs in the denominator of our **z-score** calculation.

**p-hat** = 0.65**p** = 0.60**q** = 1 - **p** = 1 - 0.60 = 0.40**n** = 30

**z** = (0.65 - 0.60) / √([0.60 * 0.40] / 30)**z** = (0.05) / √([0.24] / 30)**z** = (0.05) / √(0.008)**z** = (0.05) / (0.089)**z** = 0.56

**p-value** = 0.7123 or 71.23%

**Answer**: 71.23% is the probability of finding a sample of 30 people with a sample proportion less than 0.65.

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