The Normal Distribution

Question

A set of 1500 exam scores is normally distributed with a mean = 84 and standard deviation = 9.

Use the Empirical Rule to complete the statements below.

1. How many students scored higher than 84? 
2. 
   
3. How many students scored between 75 and 93? 
4. 
   
5. How many students scored between 66 and 102? 
6. 
   
7. How many students scored between 84 and 93? 
8. 
   
9. How many students scored higher than 75? 

Answer 1

The empirical rule a normal distribution with mean μ and standard deviation σ states that 68% of the data falls within one standard deviation [μ - σ, μ + σ], 95% percent within two standard deviations [μ - 2σ, μ + 2σ], and 99.7% within three standard deviations from the mean [μ - 3σ, μ + 3σ]. Based on that, it becomes easy to tackle the given questions:

a) The normal distribution is centered at the mean, so half of the students will have a score > 84. So 1500/2 = 750 students.

b) The interval [75,93] is the interval [μ - σ, μ + σ], so it is going to be 68% * 1500 = 1020 students.

c) The interval [66,102] is the interval [μ - 2σ, μ + 2σ], so it is going to be 95% * 1500 = 1425 students.

d) The interval [84,93] is the interval [μ, μ + σ], so it is going to be 68% / 2 * 1500 = 510 students.

e) The interval [75,+∞) is the interval [μ-σ, ∞), so it is going to be (68% / 2 + 50%) * 1500 = 1260 students.

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