Questions: Homework Question 5, 5.2.17 HW Score: 40%, 4 of 10 points Part 1 of 2 Points: 0 of 1 Save Use the normal distribution of SAT critical reading scores for which the mean is 515 and the standard deviation is 121. 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 525? 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 675 (Round to two decimal places as needed.)

Solution Steps

To find the percentage of SAT verbal scores less than 675, we first calculate the Z-score using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

where \(X = 675\), \(\mu = 515\), and \(\sigma = 121\).

Calculating the Z-score:

\[ Z_{end} = \frac{675 - 515}{121} \approx 1.3223 \]

Using the Z-score, we find the probability \(P\) that a score is less than 675:

\[ P = \Phi(Z_{end}) - \Phi(-\infty) = \Phi(1.3223) - 0 = 0.907 \]

Thus, the percentage of SAT verbal scores less than 675 is:

\[ P \times 100 = 90.70\% \]

Next, we calculate the Z-score for 525:

\[ Z_{start} = \frac{525 - 515}{121} \approx 0.0826 \]

Now, we find the probability \(P\) that a score is greater than 525:

\[ P = \Phi(\infty) - \Phi(Z_{start}) = 1 - \Phi(0.0826) \approx 0.4671 \]

To find the expected number of scores greater than 525 out of 1000, we multiply the probability by 1000:

\[ \text{Expected Count} = P \times 1000 = 0.4671 \times 1000 \approx 467.1 \approx 467 \]

Final Answer