Z-scores with sample mean

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?

ClueInsight
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!
Created with Statistics Kingdom

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.

Activate AutoScroll

You haven't unlocked all of ISA 225 yet...

Unlock our 100 concept breakdowns & 57 practice problems with guided solution walkthroughs!