Question

On a nationwide test taken by high school students, the mean score was 51 and the standard deviation was 11

The scores were normally distributed. Complete the following statements.

(a) Approximately ?% of the students scored between 40 and 62 . (b) Approximately 95% of the students scored between ? and ?

Answer 1

Solution

Back-up Theory

If a random variable X ~ N(µ, σ²), i.e., X has Normal Distribution with mean µ and variance σ², then,

By Empirical rule, also known as 68 – 95 – 99.7 percent rule:

P{(µ - σ) ≤ X ≤ ( µ + σ)} = 0.68 or 68% ……………………………………………….(1a)

P{(µ - 2σ) ≤ X ≤ (µ + 2σ)} = 0.95 or 95% ……………………………………………….(1b)

P{( µ - 3σ) ≤ X ≤ ( µ + 3σ)} = 0.997 or 99.7% ……………………………………………….(1c)

Now to work out the solution,

Let X = test score of high school students. Then, we are given X ~ N(µ, σ²), where

µ = 51 and σ = 11……………………………………………………………………………. (2)

Part (a)

40 = (51 – 11)

     = µ - σ and

62 = 51 + 11

= µ + σ.

So, by (1a) approximate % of the students with scores between 40 and 62

= 68% Answer

Part (b)

Let t₁ and t₂ be the scores within which approximately 95% of the students scored. Then by (1b),

t₁ = (µ - 2σ) = 51 – (2 x11) = 29 and

t₂ = (µ + 2σ) = 51 + (2 x11) = 73

Thus, approximately 95% of the students scored between 29 and 73. Answer

DONE