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 percentage of the population has an IQ score less than 105?

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

We're given a mean (µ) of 100. | We're working with means here, not proportions. |

We're being asked to consider the mean from the population. | We'll be working with the population distribution, not a sampling distribution. |

IQ scores follow a standard normal distribution. | 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!

**x** is the value we choose.**µ** is the population mean.**σ **is the population standard deviation.

**Standard Error (SE)** is still occurring in the denominator of our z-score formula!

Since we're working with the population mean,

. And anything divided by 1 is itself!**n** = 1

**x** = 105**µ** = 100**σ** = 15

**z** = (105 - 100) / 15**z** = (5) / 15**z** = 0.33

**p-value** = 0.6293 or 62.93%

**Answer**: 62.93% of the population has an IQ score less than 105.

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