Question \#6 (16 points)

Student scores on Professor Combs' Stats final exam are normally distributed with a mean of 75 and a standard deviation of 6.8

Find the probability of the following:
$* *$ (use 4 decimal places) ${ }^{* *}$
a.) The probability that one student chosen at random scores above an 80 .
b.) The probability that 10 students chosen at random have a mean score above an 80 .
c.) The probability that one student zhosen at random scores between a 70 and an 80 .
d.) The probability that 10 students chosen at random have a mean score between a 70 and an 80 .

Step 1 ：Convert the score of 80 to a z-score using the formula \(Z = \frac{X - \mu}{\sigma}\), where \(X\) is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. So, \(Z = \frac{80 - 75}{6.8} = 0.7353\).

Step 2 ：Look up this z-score in a standard normal distribution table or use a calculator to find the probability. The table or calculator gives the probability for scores up to the z-score, so subtract this from 1 to find the probability for scores above the z-score. So, \(P(X > 80) = 1 - P(Z < 0.7353) = 1 - 0.7681 = 0.2319\).

Step 3 ：\(\boxed{0.2319}\) is the probability that one student chosen at random scores above an 80.

Step 4 ：When dealing with the mean score of a sample, the standard deviation is divided by the square root of the sample size (n). This is known as the standard error (SE). So, \(SE = \frac{\sigma}{\sqrt{n}} = \frac{6.8}{\sqrt{10}} = 2.15\).

Step 5 ：Find the z-score for a mean score of 80: \(Z = \frac{80 - 75}{2.15} = 2.3256\).

Step 6 ：Find the probability: \(P(\overline{X} > 80) = 1 - P(Z < 2.3256) = 1 - 0.9898 = 0.0102\).

Step 7 ：\(\boxed{0.0102}\) is the probability that 10 students chosen at random have a mean score above an 80.

Step 8 ：Find the z-scores for 70 and 80: \(Z1 = \frac{70 - 75}{6.8} = -0.7353\) and \(Z2 = \frac{80 - 75}{6.8} = 0.7353\).

Step 9 ：Find the probability: \(P(70 < X < 80) = P(Z1 < Z < Z2) = P(Z < 0.7353) - P(Z < -0.7353) = 0.7681 - 0.2310 = 0.5371\).

Step 10 ：\(\boxed{0.5371}\) is the probability that one student chosen at random scores between a 70 and an 80.

Step 11 ：Find the z-scores for mean scores of 70 and 80: \(Z1 = \frac{70 - 75}{2.15} = -2.3256\) and \(Z2 = \frac{80 - 75}{2.15} = 2.3256\).

Step 12 ：Find the probability: \(P(70 < \overline{X} < 80) = P(Z1 < Z < Z2) = P(Z < 2.3256) - P(Z < -2.3256) = 0.9898 - 0.0102 = 0.9796\).

Step 13 ：\(\boxed{0.9796}\) is the probability that 10 students chosen at random have a mean score between a 70 and an 80.