Question

A snack company makes packages of grapes and honeydew slices. The weight of an individual package of
grapes, G, is approximately Normally distributed with a mean of 3.25 ounces and a standard deviation of 0.91
ounces. The weight of an individual package of honeydew slices, H, is approximately Normally distributed with a
mean of 4.51 ounces and a standard deviation of 2.02 ounces. Assume G and H are independent random
variables. Let D=G-H. What is the probability that a randomly selected package of grapes weighs more than a
randomly selected package of honeydew slices?
O 0.127
O 0.230
0.284
0.334

Answer 1

Using the normal distribution, it is found that there is a 0.284 = 28.4% probability that a randomly selected package of grapes weighs more than a  randomly selected package of honeydew slices.

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

• It measures how many standard deviations the measure is from the mean.  

• After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.

• When two normal variables are subtracted, the mean is the subtraction of the means, while the standard deviation is the square root of the sum of the variances.

In this problem, for each variable, $\mu_G = 3.25, \sigma_G = 0.91, \mu_H = 4.51, \sigma_H = 2.02$

For the subtraction, we have that:

$\mu_D = \mu_G - \mu_H = 3.25 - 4.51 = -1.26$

$\sigma_D = \sqrt{\sigma_G^2 + \sigma_H^2} = \sqrt{0.91^2 + 2.02^2} = 2.2155$

The probability that a randomly selected package of grapes weighs more than a  randomly selected package of honeydew slices is P(D > 0), which is 1 subtracted by the p-value of Z when X = 0.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{0 - (-1.26)}{2.2155}$

$Z = 0.57$

$Z = 0.57$ has a p-value of 0.7157.

1 - 0.715 = 0.284

0.284 = 28.4% probability that a randomly selected package of grapes weighs more than a  randomly selected package of honeydew slices.

A similar problem is given at https://brainly.com/question/24663213