Normal Curve Problems Using the Empirical Rule
2. Perform calculations (show work)
Assume IQ is normally distributed with mean 100 points and standard deviation 15 points. What percent of test takers score at least 70 points? Use the sketch area to show your work and enter your answer below.
Why is the answer 97.5?
To solve this problem, we need to use the Empirical Rule, which states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation from the mean
- Approximately 95% of the data falls within two standard deviations from the mean
- Approximately 99.7% of the data falls within three standard deviations from the mean
First, we need to find the z-score for a score of 70. The z-score formula is:
z = (X - μ) / σ
where:
X = score of interest (70)
μ = mean (100)
σ = standard deviation (15)
Plugging in the values, we get:
z = (70 - 100) / 15
z = -30 / 15
z = -2
A z-score of -2 corresponds to the 2nd standard deviation below the mean. In other words, only 2.5% of the data falls below this score. Since we are interested in finding the percentage of test takers scoring at least 70 points, we need to find the percentage above the score of 70.
As per the Empirical Rule, 95% of the data falls within two standard deviations from the mean. Therefore, the percentage of test takers scoring at least 70 points is:
100% - 2.5% = 97.5%
So, the answer is 97.5% because only 2.5% of the data falls below a score of 70, and 95% falls within two standard deviations above the mean.