The distribution of the SAT scores in math for an incoming class of business students has a mean of 590 and Standard deviation of 22. Assume that the scores are normally distributed.

a. Find the probability that an individual's SAT score is less than 550.
b. Find the probability than an individual's SAT score is between 550 and 600.
c. Find the probability that an individual's SAT score is greater than 600.
d. What percentage of students will have scored better than 700?
e. Find the standardized values for the students scoring 550, 600, 650, and 700 on the test.

Solution

Given that mean $\mu=590$ and standard deviation $=\sigma = 22$.

a. The probability that an individual's SAT score is less than 550 is

$$ \begin{aligned} P(X < 550) &= P\bigg(\frac{X-\mu}{\sigma} < \frac{550-590}{22}\bigg)\\ &= P\bigg(Z < -1.818\bigg)\\ &=0.0345 \end{aligned} $$

Normal Distribution
Normal Distribution

b. The probability than an individual's SAT score is between 550 and 600 is

$$ \begin{aligned} P(550 < X< 600) &= P(X < 600) - P(X < 550)\\ &= P\bigg(\frac{X-\mu}{\sigma} < \frac{600-590}{22}\bigg)-P\bigg(\frac{X-\mu}{\sigma} < \frac{550-590}{22}\bigg)\\ &=P\bigg(Z < 0.455\bigg)-P\bigg(Z < -1.818\bigg)\\ &=0.6753-0.0345\\ &= 0.6408 \end{aligned} $$

Normal Distribution
Normal Distribution

c. The probability that an individual's SAT score is greater than 600 is

$$ \begin{aligned} P(X > 600) &= 1-P(X\leq600)\\ &= 1- P\bigg(\frac{X-\mu}{\sigma} \leq \frac{600-590}{22}\bigg)\\ &=1- P\bigg(Z \leq 0.455\bigg)\\ &=1-0.6753\\ &= 0.3247 \end{aligned} $$

Normal Distribution
Normal Distribution

d. The probability that the students scored better than 700 is

$$ \begin{aligned} P(X > 700) &= 1-P(X\leq700)\\ &= 1- P\bigg(\frac{X-\mu}{\sigma} \leq \frac{700-590}{22}\bigg)\\ &=1- P\bigg(Z \leq 5\bigg)\\ &=1-1\\ &= 0 \end{aligned} $$

Normal Distribution
Normal Distribution

0% of the students will have scored better than 700.

e. The standardized values:

The standardized values for the students scoring 550 is

$$ \begin{aligned} Z& =\frac{550-590}{22}\\ &= -1.818 \end{aligned} $$

The standardized values for the students scoring 600 is

$$ \begin{aligned} Z& =\frac{600-590}{22}\\ &= 0.455 \end{aligned} $$

The standardized values for the students scoring 650 is

$$ \begin{aligned} Z& =\frac{650-590}{22}\\ &= 2.727 \end{aligned} $$

The standardized values for the students scoring 700 is

$$ \begin{aligned} Z& =\frac{700-590}{22}\\ &= 5 \end{aligned} $$