Question

test scores are normally distributed with a mean of 76 and a standard deviation of 10.15. in a group of 230 tests, how many students score above 96.16. in a group of 230 tests, how many students score below 66.17. in a group of 230 tests, how many students score within one standard deviation of the mean?

Explanation:

Step1: Calculate z - score

The z - score formula is \(z=\frac{x-\mu}{\sigma}\).
For \(x = 96\), \(\mu=76\), \(\sigma = 10\), then \(z=\frac{96 - 76}{10}=2\)
For \(x = 66\), \(z=\frac{66 - 76}{10}=- 1\)
For one - standard - deviation within the mean (\(z=-1\) to \(z = 1\))

Step2: Use the empirical rule (68 - 95 - 99.7 rule)

The empirical rule states that approximately 68% of the data lies within one - standard - deviation of the mean, 95% within two - standard - deviations, and 99.7% within three - standard - deviations.
For \(z = 2\), using the standard normal distribution table or the empirical rule (for \(z = 2\), the area to the right of \(z = 2\) is \(\frac{1 - 0.95}{2}=0.025\))
For \(z=-1\), the area to the left of \(z=-1\) is \(\frac{1 - 0.68}{2}=0.16\)

Step3: Calculate the number of students

Total number of students \(n = 230\)
For problem 15: Number of students with \(x>96\): \(n_1=230\times0.025 = 5.75\approx6\)
For problem 16: Number of students with \(x<66\): \(n_2=230\times0.16=36.8\approx37\)
For problem 17: Number of students within one - standard - deviation (\(z=-1\) to \(z = 1\)): \(n_3=230\times0.68 = 156.4\approx156\)

Answer:

1. Approximately 6 students score above 96.
2. Approximately 37 students score below 66.
3. Approximately 156 students score within one standard deviation of the mean.