An appliance dealer sells three different models of upright freezers having 13.3, 16.1, and 19.3 cubic feet of
storage space. Consider the random variable x = Amount of storage space purchased by the next customer
to buy a freezer. Suppose that x has the following probability distribution.
x p(x)
13.3 0.2
16.1 0.5
19.3 0.3
(a) Calculate the mean and standard deviation of x (in cubic feet). (Hint: See Example 6.15. Round your
standard deviation to four decimal places.)
(b) Give an interpretation of the mean and standard deviation of x in the context of observing the
outcomes of many purchases.
The mean represents the long run average distribution of freezers sold by this particular
appliance dealer. The standard deviation represents a typical deviation in how much the
distribution in freezers purchased deviates from the mean.
The mean represents the long run average number of freezers sold by this particular
appliance dealer. The standard deviation represents a typical deviation in how many freezers
purchased deviates from the mean.
The mean represents the long run average cost of freezers sold by this particular appliance
dealer. The standard deviation represents a typical deviation in how much is spent on
freezers purchased deviates from the mean.
The mean represents the long run average type of freezer sold by this particular appliance
dealer. The standard deviation represents a typical deviation in what type of freezer
purchased deviates from the mean.
The mean represents the long run average storage space of freezers sold by this particular
appliance dealer. The standard deviation represents a typical deviation in how much the
storage space in freezers purchased deviates from the mean.
Guide On Rating System
Vote
(a) To calculate the mean of x, we multiply each value of x by its respective probability and sum them up:
mean = (13.3 * 0.2) + (16.1 * 0.5) + (19.3 * 0.3)
= 2.66 + 8.05 + 5.79
= 16.5 cubic feet
To calculate the standard deviation of x, we need to calculate the variance first. The variance is calculated by squaring the difference between each value of x and the mean, multiplying it by its respective probability, and summing them up:
variance = (13.3 - 16.5)^2 * 0.2 + (16.1 - 16.5)^2 * 0.5 + (19.3 - 16.5)^2 * 0.3
= 4.84 * 0.2 + 0.16 * 0.5 + 8.41 * 0.3
= 0.968 + 0.08 + 2.523
= 3.571 cubic feet^2
The standard deviation is the square root of the variance:
standard deviation = √(3.571)
≈ 1.889 cubic feet (rounded to four decimal places)
(b) The mean represents the long run average storage space of freezers sold by this particular appliance dealer. This means that if many customers purchase freezers from this dealer, the average amount of storage space they will get is approximately 16.5 cubic feet.
The standard deviation represents a typical deviation in how much the storage space in freezers purchased deviates from the mean. In other words, it measures the average amount by which the storage space of freezers purchased by customers differs from the long run average of 16.5 cubic feet. This indicates the variability in the sizes of freezers purchased by customers.
Got a question with my answer?
Cheapmath.com is completely free!
Asking question is free.
Getting answers is free.
+99