In a normal distribution with mean mu = 70 and standard deviation sigma = 8, what percentage of the population lies between 54 and 86 according to the Empirical Rule (68-95-99.7)?
Answer and explanation
Correct answer: B. Approximately 95.4%
- Step 1: Calculate z-scores for x1 = 54 and x2 = 86:
- Step 2: z1 = (54 - 70) / 8 = -16 / 8 = -2.
- Step 3: z2 = (86 - 70) / 8 = +16 / 8 = +2.
- Step 4: The range represents values within 2 standard deviations of the mean (mu +- 2 sigma).
- Step 5: According to the Empirical Rule, approximately 95.4% of the distribution lies within 2 standard deviations.
Answer
Approximately 95.4%.
Why not the others: 'Approximately 68.2%' is within 1 standard deviation (62 to 78); 'Approximately 99.7%' is within 3 standard deviations (46 to 94); 'Approximately 50.0%' is the probability above or below the mean.
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