Question

A grocer purchases pumpkins from two producers, M and N. Producer M provides pumpkins from organic farms. The distribution of the diameters of the pumpkins from Producer M is approximately normal with mean 133 millimeters (mm) and standard deviation 5 mm.

a. For a pumpkin selected at random from Producer M, what is the probability that the pumpkins will have a diameter bigger than 137 mm? Producer N provides pumpkins from nonorganic farms. The probability is 0.8413 that a pumpkin selected at random from Producer N will have a diameter greater than 137 mm. For all the pumpkins at the grocery store, 30 percent of the pumpkins are provided by Producer M and 70 percent are provided by Producer N.
b. For a pumpkin selected at random from the grocery store, what is the probability that the pumpkin will have a diameter greater than 137 mm?
c. Given that a pumpkin selected at random from the grocery store has a diameter greater than 137 mm, what is the probability that the pumpkin will be from Producer M?

Answer 1

The probability that a pumpkin selected at random from Producer M will have a diameter bigger than 137 mm is 0.2119.

Step-by-step explanation:

a. To find the probability that a pumpkin selected at random from Producer M will have a diameter bigger than 137 mm, we need to find the area under the normal distribution curve to the right of 137 mm. To calculate this, we can use the z-score formula:

z = (x - mean) / standard deviation

where x is the value we want to find the probability for, mean is the mean diameter, and standard deviation is the standard deviation of the diameters. Plugging in the values, we get:

z = (137 - 133) / 5 = 0.8

Using a standard normal distribution table or a calculator, we can find that the probability of a z-score greater than 0.8 is approximately 0.2119. Therefore, the probability that a pumpkin from Producer M will have a diameter bigger than 137 mm is 0.2119.