Question

Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.552.55 and a standard deviation of 0.440.44. Using the empirical rule, what percentage of the students have grade point averages that are between 1.231.23 and 3.873.87?

Answer 1

Given, $\mu$ = 2.55 , $\sigma$ = 0.44

We have to calculate P(1.23 < X < 3.87) = ?

We can write 3.87 in terms of $\mu$ and $\sigma$ as

3.87 = 2.55 + 3 * 0.44 = $\mu$ + 3 * $\sigma$

That is 3.87 is 3 standard deviation above the mean.

Similarly,

1.23 = 2.55 - 3 * 0.44 = $\mu$ - 3 * $\sigma$

That is 1.23 is 3 standard deviation below the mean.

Using empirical (68-95-99.7) rule,

Approximately, 99.7% of the data falls in 3 standard deviation of the mean.

Therefore,

(1.23 < X < 3.87) = 99.7%