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 (Round to four decimahslaces as needed.)

Step 1 ：The problem is asking for the probability that a randomly selected carton has a weight greater than 8.14 ounces. This is a question about the normal distribution. The normal distribution is defined by two parameters: the mean and the standard deviation. In this case, the mean is 8 ounces and the standard deviation is 0.4 ounce.

Step 2 ：To find the probability that a randomly selected carton has a weight greater than 8.14 ounces, we need to standardize the value 8.14. This is done by subtracting the mean from the value and dividing by the standard deviation. This will give us a z-score, which we can then use to find the probability.

Step 3 ：Calculate the z-score using the formula \(z = \frac{value - mean}{std\_dev}\). Substituting the given values, we get \(z = \frac{8.14 - 8}{0.4} = 0.35\).

Step 4 ：Using the z-score, we can find the probability from the standard normal distribution. The probability that a randomly selected carton has a weight greater than 8.14 ounces is approximately 0.3632.

Step 5 ：Final Answer: The probability that a randomly selected carton has a weight greater than 8.14 ounces is \(\boxed{0.3632}\).