Question

A grocery store has an express line for customers purchasing at most five items. Let x be the number of items purchased by a randomly selected customer using this line. Give examples of two different assignments of probabilities such that the resulting distributions have the same mean but quite different standard deviations.

Answer 1

The question involves creating two probability distributions for the number of items a customer purchases in a grocery store express line, with the same mean but different standard deviations. Distribution 1 spreads probability evenly, resulting in a smaller standard deviation compared to Distribution 2, which has a concentration around a central value causing a larger standard deviation.

Step-by-step explanation:

The question is about creating two different probability distributions for the number of items a customer purchases at an express line of a grocery store, where the random variable X represents the number of items. Both distributions should have the same mean but different standard deviations. We will consider that the express line is for customers purchasing at most five items.

Let's assume the following two distributions for the number of items (x):

• Distribution 1:

P(X=1) = 0.2, P(X=2) = 0.2, P(X=3) = 0.2, P(X=4) = 0.2, P(X=5) = 0.2

• Distribution 2:

P(X=1) = 0.1, P(X=2) = 0.1, P(X=3) = 0.6, P(X=4) = 0.1, P(X=5) = 0.1

Both distributions have a mean of 3 items (since 1(0.2)+2(0.2)+3(0.2)+4(0.2)+5(0.2) = 1(0.1)+2(0.1)+3(0.6)+4(0.1)+5(0.1) = 3). However, the standard deviation of the first distribution is smaller than that of the second because the probabilities are evenly spread out in Distribution 1, while Distribution 2 has a heavier concentration at 3 items, giving it a larger variance and thus a larger standard deviation.