Question

use the normal distribution of sat critical reading scores for which the mean is 500 and the standard deviation is 115. assume the variable x is normally distributed. (a) what percent of the sat verbal scores are less than 675? (b) if 1000 sat verbal scores are randomly selected, about how many would you expect to be greater than 575? click to view page 1 of the standard normal tables. click to view page 2 of the standard normal table. (a) approximately % of the sat verbal scores are less than 675 (round to two decimal places as needed.)

Explanation:

Step1: Calculate z - score for part (a)

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 500$, $\sigma=115$ and $x = 675$. So $z=\frac{675 - 500}{115}=\frac{175}{115}\approx1.52$.

Step2: Find the proportion for part (a)

Using the standard - normal table, the proportion of values to the left of $z = 1.52$ is approximately $0.9357$. To convert this to a percentage, we multiply by 100. So the percentage of SAT verbal scores less than 675 is $0.9357\times100 = 93.57\%$.

Step3: Calculate z - score for part (b)

For $x = 575$, using the z - score formula $z=\frac{x-\mu}{\sigma}$, we have $z=\frac{575 - 500}{115}=\frac{75}{115}\approx0.65$.

Step4: Find the proportion for part (b)

The proportion of values to the left of $z = 0.65$ from the standard - normal table is approximately $0.7422$. The proportion of values greater than $z = 0.65$ is $1 - 0.7422=0.2578$.

Step5: Calculate the expected number for part (b)

If $n = 1000$ SAT verbal scores are selected, the expected number of scores greater than 575 is $n\times(1 - P(Z<0.65))=1000\times0.2578 = 257.8\approx258$.

Answer:

(a) $93.57$
(b) $258$