Use the normal distribution of SAT critical reading scores for which the mean is 509 and the standard deviation is 119. Assume the variable $x$ is normally distributed.

(a) What percent of the SAT verbal scores are less than 550?

(b) If 1000 SAT verbal scores are randomly selected, about how many would you expect to be greater than 525?

#### Solution

Here $X : $ SAT critical reasoning score. And $X$ is normally distributed with mean `$\mu = 509$`

and standard deviation `$\sigma = 119$`

. That is, `$X\sim N(509, 119^2)$`

.

(a) First let is find the probability that the SAT scores are less than 550.

` $$ \begin{eqnarray*} P(X < 550) & = & P\bigg(\frac{X-\mu}{\sigma} < \frac{550-509}{119}\bigg) \\ & = & P\big(Z < \frac{550-509}{119}\big)\\ & = & P(Z < 0.3445378)\\ & = & P(Z< 0.34) \\ & = & 0.6348 \end{eqnarray*} $$ `

So 63.68 % i.e. 64% of the SAT verbal scores are less than 550.

(b) Here $N =1000$.

The probability that SAT verbal scores are greater than 525 is

` $$ \begin{aligned} P(X\geq 525) & = 1-P\bigg(\frac{X-\mu}{\sigma} \leq \frac{525-509}{119} \bigg)\\ & = 1-P\big(Z\leq 0.13 \big)\\ &= 1- 0.5535\\ &=0.4465 \end{aligned} $$ `

So expected number of SAT verbal scores greater than 525 is `$= N* P(X > 525) = 1000* 0.4465 = 446.5 = 447$`

.

Hence if 1000 SAT verbal scores are randomly selected, about 447 would be expected to greater than 525.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators