ACT (Enhanced)MathematicsMedium
A data scientist is analyzing a dataset of customer ages. The ages are normally distributed with a mean of 35 years and a standard deviation of 5 years. Approximately what percentage of customers are between 30 and 40 years old?
- A99.7%
- B95%
- C68%
- D34%
Show answer & explanationAnswer & explanation
Correct answer: C. 68%
The range of 30 to 40 years old is one standard deviation below the mean (35-5=30) and one standard deviation above the mean (35+5=40). According to the empirical rule (68-95-99.7 rule) for normal distributions, approximately 68% of data falls within one standard deviation of the mean.
Why the other options are wrong
- A. This percentage covers three standard deviations from the mean (35 ± 3*5 = 20 to 50).
- B. This percentage covers two standard deviations from the mean (35 ± 2*5 = 25 to 45).
- D. This is the approximate percentage between the mean and one standard deviation above (or below) the mean.
Empirical Rule (68-95-99.7)
A statistical rule stating that for a normal distribution, approximately 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations of the mean.
- Applies specifically to normal (bell-shaped) distributions.
- Useful for quick estimations of data spread.
- Also known as the 3-sigma rule.
Memory trick: Bell's Rule: 1-sigma is 68, 2-sigma is 95, 3-sigma is 99.7!