Use the normal distribution of SAT critical reading scores for which the mean is 505 and the standard deviation is 114. 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 550 ?
Click to view page 1 of the standard normal table.
Click to view page 2 of the standard normal table.
(a) Approximately $93.20\mathrm{\%}$ of the SAT verbal scores are less than 675
(Round to two decimal places as needed.)
(b) You would expect that approximately
SAT verbal scores would be greater than 550 . (Round to the nearest whole number as needed.)

Step 1 ：Given that the mean ($\mu$) is 505 and the standard deviation ($\sigma$) is 114, we want to find the percentage of SAT verbal scores less than 675 and the expected number of scores greater than 550 out of 1000.

Step 2 ：For part (a), we first calculate the z-score for 675. The z-score is calculated as $z=\frac{x-\mu }{\sigma }$. Substituting the given values, we get $z_a=\frac{675-505}{114}=1.4912280701754386$.

Step 3 ：We then look up this z-score in the standard normal table to find the corresponding percentile. The percentile for $z_a=1.4912280701754386$ is approximately 93.20\%.

Step 4 ：$\boxed{{\displaystyle \text{Therefore, approximately 93.20\% of the SAT verbal scores are less than 675.}}}$

Step 5 ：For part (b), we calculate the z-score for 550 in the same way as in part (a). We get $z_b=\frac{550-505}{114}=0.39473684210526316$.

Step 6 ：We look up this z-score in the standard normal table to find the corresponding percentile. The percentile for $z_b=0.39473684210526316$ is approximately 65.35\%.

Step 7 ：This means that approximately 65.35\% of the scores are less than 550. To find the proportion of scores that are greater than 550, we subtract this value from 1. We get $1-0.6535=0.3465$.

Step 8 ：To find the expected number of scores greater than 550 out of 1000, we multiply this proportion by 1000. We get $0.3465\times 1000=346.51855503363174$.

Step 9 ：$\boxed{{\displaystyle \text{Therefore, we would expect that approximately 347 SAT verbal scores would be greater than 550.}}}$