use the normal distribution of sat critical reading scores for which the mean is 514 and the standard…

use the normal distribution of sat critical reading scores for which the mean is 514 and the standard deviation is 124. assume the variable x is normally distributed. (a) what percent of the sat verbal scores are less than 625? (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 table click to view page 2 of the standard normal table. (a) approximately % of the sat verbal scores are less than 625 (round to two decimal places as needed.)

use the normal distribution of sat critical reading scores for which the mean is 514 and the standard deviation is 124. assume the variable x is normally distributed. (a) what percent of the sat verbal scores are less than 625? (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 table click to view page 2 of the standard normal table. (a) approximately % of the sat verbal scores are less than 625 (round to two decimal places as needed.)

Answer

Explanation:

Step1: Calculate z - score for part (a)

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 514$, $\sigma=124$, and $x = 625$. So $z=\frac{625 - 514}{124}=\frac{111}{124}\approx0.8952$.

Step2: Find the percentile for part (a)

Using the standard - normal table, the area to the left of $z = 0.8952$ (which represents the percentage of scores less than 625) is approximately $0.8133$ or $81.33%$.

Step3: Calculate z - score for part (b)

For $x = 575$, $z=\frac{575 - 514}{124}=\frac{61}{124}\approx0.4919$.

Step4: Find the area to the right for part (b)

The area to the left of $z = 0.4919$ from the standard - normal table is approximately $0.688$. The area to the right (scores greater than 575) is $1 - 0.688 = 0.312$.

Step5: Calculate the expected number for part (b)

If $n = 1000$ scores are selected, the expected number of scores greater than 575 is $n\times(1 - P(Z\leq0.4919))=1000\times0.312 = 312$.

Answer:

(a) $81.33$ (b) $312$