The average price of a college math textbook is $\$ 161$ and the standard deviation is $\$ 30$. Suppose that 18 textbooks are randomly chosen. Round all answers to 4 decimal places where possible. For the group of 18 , find the probability that the average price is between $\$ 160$ and \$171. Find the third quartile for the average textbook price for this sample size.

Step 1 ：Given that the average price of a college math textbook is \$161 with a standard deviation of \$30, we are asked to find the probability that the average price of 18 randomly chosen textbooks is between \$160 and \$171.

Step 2 ：First, we calculate the standard deviation for the average price of 18 textbooks using the formula for the standard deviation of the mean. This gives us a standard deviation of approximately \(7.0710678118654755\).

Step 3 ：Next, we calculate the z-scores for \$160 and \$171 using the z-score formula. The z-score for \$160 is approximately \(-0.1414213562373095\) and the z-score for \$171 is approximately \(1.414213562373095\).

Step 4 ：We then use the standard normal distribution to find the probabilities corresponding to these z-scores. The probability corresponding to the z-score for \$160 is approximately \(0.4437685419908576\) and the probability corresponding to the z-score for \$171 is approximately \(0.9213503964748574\).

Step 5 ：The probability that the average price is between \$160 and \$171 is the difference between these two probabilities, which is approximately \(0.4775818544839998\).

Step 6 ：For the second part of the question, we are asked to find the third quartile for the average textbook price for this sample size. The third quartile is the value below which 75% of the data fall.

Step 7 ：We use the z-score corresponding to the 75th percentile in the standard normal distribution, which is approximately \(0.6744897501960817\).

Step 8 ：We then convert this z-score back to a price using the mean and standard deviation of the average price. This gives us a third quartile of approximately \$165.7693627620447.

Step 9 ：Final Answer: The probability that the average price of 18 randomly chosen textbooks is between \$160 and \$171 is approximately \(\boxed{0.4776}\). The third quartile for the average textbook price for this sample size is approximately \(\boxed{165.77}\).