Question

SAT verbal scores are normally distributed with a mean of 433 and a standard deviation of 90. Use the Empirical Rule to determine what percent of the scores lie between 433 and 523.

Answer 1

34% of the scores lie between 433 and 523.

Solution:

Given data:

Mean (μ) = 433

Standard deviation (σ) = 90

Empirical rule to determine the percent:

(1) About 68% of all the values lie within 1 standard deviation of the mean.

(2) About 95% of all the values lie within 2 standard deviations of the mean.

(3) About 99.7% of all the values lie within 3 standard deviations of the mean.

$Z(X)=\frac{x-\mu}{\sigma}$

$Z(433)=\frac{433-\ 433}{90}=0$

$Z(523)=\frac{523-\ 433}{90}=1$

Z lies between o and 1.

P(433 < x < 523) = P(0 < Z < 1)

μ = 433 and μ + σ = 433 + 90 = 523

Using empirical rule, about 68% of all the values lie within 1 standard deviation of the mean.

i. e. $((\mu-\sigma) \ \text{to} \ (\mu+\sigma))=68\%$

Here μ to μ + σ = $\frac{68\%}{2} =34\%$

Hence 34% of the scores lie between 433 and 523.