Z-scores with population mean

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?

ClueInsight
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!
Created with Statistics Kingdom

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

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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