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**: Given the distribution of all IQ scores has a mean (µ) of 100 and standard deviation (σ) of 15,* *what's the probability of finding a sample of 30 people with a sample mean IQ score less than 105?

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

We've got a mean (µ) of 100. | We're working with means here. |

We're being asked to find a mean from a sample of 30 people. | We'll be working with a sampling distribution, not a population distribution. |

The population distribution for IQ scores is normal, so our sampling distribution will be too. | We can utilize z-scores, not t-scores. |

What we'll do:

- Compute z-score, given our x-value of 105
- 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!

The **Standard Error (SE)** formula occurs in the denominator!

**x** = 105**µ** = 100**σ** = 15**n** = 30

**z** = (105 - 100) / [15 / √(30)]**z** = (5) / [15 / (5.48)]**z** = (5) / [2.74]**z** = 1.83

**p-value** = 0.9664 or 96.64%

**Answer**: 96.64% is the probability of finding a sample of 30 people with a sample mean IQ score less than 105.

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