ACT (Enhanced)MathematicsHard
A production manager is reviewing the efficiency of an assembly line. The number of defects per 1000 units produced is normally distributed with a mean of 15 defects and a standard deviation of 3 defects. What is the probability that a randomly selected batch of 1000 units will have between 12 and 18 defects?
- A68%
- B99.7%
- C34%
- D95%
Show answer & explanationAnswer & explanation
Correct answer: A. 68%
The range 'between 12 and 18 defects' is exactly one standard deviation (3 defects) below and one standard deviation above the mean (15 defects). According to the Empirical Rule (68-95-99.7), approximately 68% of data falls within one standard deviation of the mean in a normal distribution.
Why the other options are wrong
- B. This represents three standard deviations from the mean (e.g., between 6 and 24 defects).
- C. This represents one side of the mean (e.g., mean to +1 std dev), not the full range.
- D. This represents two standard deviations from the mean (e.g., between 9 and 21 defects).
Empirical Rule (68-95-99.7)
A rule for normal distributions stating that approximately 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean.
- Applies specifically to normal (bell-shaped) distributions.
- Provides quick estimates of probabilities.
- Standard deviations are measured from the mean.
Memory trick: Sixty-eight, ninety-five, ninety-nine point seven, for one, two, and three std devs, a data heaven!