Question

the weights of red delicious apples are approximately normally distributed with a mean of 9 ounces and a standard deviation of 0.75 ounce. an online gift store sells gift boxes containing 5 red delicious apples. at the time of packaging, 5 red delicious apples are randomly selected and packaged in a box. describe the distribution of the total weight of the 5 randomly selected apples. what is the probability that the total weight of the 5 randomly selected apples will be less than 42 ounces? the combined weight of the packing material and box in which the apples will be shipped is always 10 ounces. let w represent the weight of a complete packaged gift box, which consists of the packing material, box, and 5 randomly selected apples. what are the mean and the standard deviation of w ?

Answer 1

The distribution of the total weight of the 5 randomly selected apples is approximately normal with a mean of 45 ounces and a standard deviation of 1.32 ounces. The probability that the total weight will be less than 42 ounces is approximately 1.18%. The mean and standard deviation of the complete packaged gift box, including the packing material and box, are 55 ounces and 1.32 ounces, respectively.

Step-by-step explanation:

The distribution of the total weight of the 5 randomly selected apples can be described as approximately normally distributed, with a mean equal to the sum of the mean weights of the individual apples (5 times the mean weight of a single apple) and a standard deviation equal to the square root of the sum of the variances of the individual apples.

In this case, the mean of the total weight would be 5 times 9 ounces, which is 45 ounces, and the standard deviation would be the square root of 5 times 0.75 squared, which is approximately 1.32 ounces.

To find the probability that the total weight of the 5 randomly selected apples will be less than 42 ounces, we can use the z-score formula.

The z-score is calculated by subtracting the mean from the desired value (42 ounces) and then dividing by the standard deviation.

With a z-score of -2.27, we can use a standard normal distribution table or calculator to find the corresponding probability, which is approximately 0.0118 or 1.18%.

To find the mean and standard deviation of the complete packaged gift box, we need to add the weight of the packing material and box (10 ounces) to the total weight of the 5 randomly selected apples.

This gives us a mean of 45 + 10 = 55 ounces and a standard deviation of 1.32 ounces.

Answer 2

The total weight of the 5 randomly selected apples follows a normal distribution with a mean of 45 ounces and a standard deviation of 1.5 ounces. The probability that the total weight will be less than 42 ounces is approximately 0.0228. The mean and standard deviation of a complete packaged gift box, including the 5 apples and the packing material and box, are 55 ounces and 1.514 ounces respectively.

Step-by-step explanation:

The total weight of the 5 randomly selected red delicious apples follows a normal distribution. The mean of the total weight is the sum of the means of the individual apples, which is 5 * 9 = 45 ounces. The standard deviation of the total weight is the square root of the sum of the variances of the individual apples, which is sqrt(5 * (0.75^2)) = 1.5 ounces.

To find the probability that the total weight of the 5 randomly selected apples will be less than 42 ounces, we need to convert this value to a Z-score. The Z-score is calculated as (X - mean) / standard deviation. In this case, (42 - 45) / 1.5 = -2. The probability of a Z-score less than -2 can be looked up in a standard normal distribution table or calculated using a calculator, which is approximately 0.0228.

The mean weight of a complete packaged gift box, denoted as w, is the sum of the mean weight of the 5 apples and the weight of the packing material and box, which is 45 + 10 = 55 ounces. The standard deviation of w is the square root of the sum of the variances of the 5 apples and the variance of the packing material and box, which is sqrt((5 * (0.75^2)) + (0.12^2)) = 1.514 ounces.