Question

Please provide how the table and the software step by step. That way I can practice . Thank you so much.

Potatoes - Samples: Assume the weights of Farmer Carl's potatoes are normally distributed with a mean of 8.0 ounces and a standard deviation of 1.1 ounces. He bags his potatoes in groups of 6. You buy a bag and the total weight is 42 ounces. Here we determine how lucky or unlucky you are.

(a) What is the mean potato weight in your bag of 6? Enter your answer to 1 decimal place. ounces

b) If 6 potatoes are randomly selected, find the probability that the mean weight is less than the mean found in your bag. Round your answer to 4 decimal places.

(c) Which statement best describes your situation?

This is not particularly unusual because the mean weight in your bag is only 1 ounce below the mean of all Carl's potatoes.

This is an unusually small amount of potatoes. You are unlucky because the probability of getting a bag weighing less than yours is only about 1.3%.

You got extremely unlucky because the probability of getting a bag weighing less than yours is about 0.01%.

You got lucky with such a generous amount of potato your bag.

Answer 1

a) The total weight of the bag we get is 42 ounces. And the bag has 6 potatoes. Hence the mean potato weight in the bag is = 42/6 = 7 ounces.

b) If 6 potatoes are randomly selected, their mean will be, say .

Further, we know that, if X is normally distributed with mean and variance, then is normally distributed with mean variance,.

Hence the probability that mean weight () is less than the mean weight found in our bag(i. e. 7 ounces) is,

Where, Z is Standard Normal Variate i.e. Z ~ N(0, 1)

c)

Note that, the probability of randomly selected 6 potatoes will have a mean weight less than that of our bag (i.e. 7 ounces) is only 0.013. i.e. 1.3%.

Hence, the correct statement is -  This is an unusually small amount of potatoes. You are unlucky because the probability of getting a bag weighing less than yours is only about 1.3%.