Question

The weights of ice cream cartons are normally distributed with a mean weight of 12 ounces and a standard deviation of 0.6 ounce.

​(a) What is the probability that a randomly selected carton has a weight greater than 12.22 ​ounces?
​(b) A sample of 36 cartons is randomly selected. What is the probability that their mean weight is greater than 12.22 ​ounces?

Answer 1

(a) 0.3594 (or 35.94%)

(b) 0.0139 (or 1.39%)

Explanation:

(a) Probability that a randomly selected carton has a weight greater than 12.22 ounces:

Calculate the Z-score using the formula Z = (X - mu) / sigma:

Z = (12.22 - 12) / 0.6 ≈ (0.22 / 0.6) ≈ 0.3667

Now, use the Z-table to find P(Z > 0.3667), which is approximately 1 - 0.6406 ≈ 0.3594 or 35.94%.

(b) Probability that the mean weight of a sample of 36 cartons is greater than 12.22 ounces:

Calculate the standard error (SE) and the Z-score for the sample mean:

SE = sigma / sqrt(n) = 0.6 / sqrt(36) = 0.1

Z = (X - mu) / SE = (12.22 - 12) / 0.1 = (0.22 / 0.1) = 2.2

Now, use the Z-table to find P(Z > 2.2), which is approximately 0.0139 or 1.39%.