The weights of ice cream cartons are normally distributed with a mean weight of 8 ounces and a standard deviation of 0.4 ounce. (a) What is the probability that a randomly selected carton has a weight greater than 8.14 ounces? (b) A sample of 36 cartons is randomly selected. What is the probability that their mean weight is greater than 8.14 ounces? (a) The probability is 0.3632 (Round to four decimal places as needed.) (b) The probability is (Round to four decimahilaces as needed.)

Step 1 ：Given that the weights of ice cream cartons are normally distributed with a mean weight of 8 ounces and a standard deviation of 0.4 ounce.

Step 2 ：For part (a), we need to calculate the z-score for the weight 8.14 ounces. The z-score is a measure of how many standard deviations an element is from the mean. We can calculate it using the formula: \(z = \frac{X - \mu}{\sigma}\) where X is the value we are interested in, \(\mu\) is the mean and \(\sigma\) is the standard deviation. Substituting the given values, we get \(z = \frac{8.14 - 8}{0.4} = 0.35\).

Step 3 ：Once we have the z-score, we can use a z-table or a statistical function to find the probability that a randomly selected carton has a weight greater than 8.14 ounces. The probability is approximately 0.3632.

Step 4 ：For part (b), we need to calculate the probability that the mean weight of a sample of 36 cartons is greater than 8.14 ounces. The standard deviation of the sample mean is given by \(\frac{\sigma}{\sqrt{n}}\), where n is the sample size. Substituting the given values, we get \(\frac{0.4}{\sqrt{36}} = 0.0667\).

Step 5 ：We can then calculate the z-score for the sample mean using the formula: \(z = \frac{X - \mu}{\sigma}\). Substituting the given values, we get \(z = \frac{8.14 - 8}{0.0667} = 2.1\).

Step 6 ：Using a z-table or a statistical function, we find the probability that the mean weight of a sample of 36 cartons is greater than 8.14 ounces. The probability is approximately 0.0179.

Step 7 ：Final Answer: (a) The probability that a randomly selected carton has a weight greater than 8.14 ounces is approximately \(\boxed{0.3632}\). (b) The probability that the mean weight of a sample of 36 cartons is greater than 8.14 ounces is approximately \(\boxed{0.0179}\).