T-scores with sample mean
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Scenario: At Crammer Nation University, the brothers of Sigma Apple Pi fraternity get an average of 25.2 Tinder matches per day. You take a sample of 35 brothers, resulting in a sample standard deviation of 2.7 matches. What's the probability that the sample mean will be above 25.9?
| Clue | Insight |
|---|---|
| Sigma Apple Pi fraternity get an average of 25.2 matches. | This is a mean, not a proportion. |
| We're assessing from a sample of 35 brothers. | We'll be using a sampling distribution, not the population distribution. |
| We only have the sample standard deviation of 2.7, not the population standard deviation. | We'll settle for a t-score, we can't use a z-score. |
What we'll do:
- Compute t-score, given our x-value of 105
- Find the area under the t-distribution to the right of our t-score (seen in green below)...
- ...which will be equal to the p-value corresponding to our t-score!
x-bar is the sample mean.
µ is the population mean.
s is the sample standard deviation.
n is the sample size.
The Standard Error (SE) formula occurs in the denominator! (Remember, we're substituting sample standard deviation (s) for population standard deviation (σ).)
x-bar = 25.9
µ = 25.2
s = 2.7
n = 35
t = (25.9 - 25.2) / [2.7 / √(35)]
t = (0.7) / [2.7 / (5.916)]
t = (0.7) / [0.456]
t = 1.535
df = n - 1
df = 35 - 1
df = 34
0.05 < p-value < 0.10
Answer: The probability that a sample of 35 brothers from Sigma Apple Pi will have a mean above 25.9 is between 5 - 10%.