Question

In the book Advanced Managerial Accounting, Robert P. Magee discusses monitoring cost variances. A cost variance is the difference between a budgeted cost and an actual cost. Magee describes the following situation:

Michael Bitner has responsibility for control of two manufacturing processes. Every week he receives a cost variance report for each of the two processes, broken down by labor costs, materials costs, and so on. One of the two processes, which we'll call process A , involves a stable, easily controlled production process with a little fluctuation in variances. Process B involves more random events: the equipment is more sensitive and prone to breakdown, the raw material prices fluctuate more, and so on.

     "It seems like I'm spending more of my time with process B than with process A," says Michael Bitner. "Yet I know that the probability of an inefficiency developing and the expected costs of inefficiencies are the same for the two processes. It's just the magnitude of random fluctuations that differs between the two, as you can see in the information below."

     "At present, I investigate variances if they exceed $2,931, regardless of whether it was process A or B. I suspect that such a policy is not the most efficient. I should probably set a higher limit for process B."

The means and standard deviations of the cost variances of processes A and B, when these processes are in control, are as follows: (Round probability answers to 4 decimal places.):

	Process A	Process B
Mean cost variance (in control)	$ 0	$ 0
Standard deviation of cost variance (in control)	$5,271	$10,270

Furthermore, the means and standard deviations of the cost variances of processes A and B, when these processes are out of control, are as follows:

	Process A	Process B
Mean cost variance (out of control)	$7,400	$ 7,381
Standard deviation of cost variance (out of control)	$5,271	$10,270

   

(a) Recall that the current policy is to investigate a cost variance if it exceeds $2,931 for either process. Assume that cost variances are normally distributed and that both Process A and Process B cost variances are in control. Find the probability that a cost variance for Process A will be investigated. Find the probability that a cost variance for Process B will be investigated. Which in-control process will be investigated more often.

Process A
Process B

(Click to select)Process AProcess B is investigated more often

(b) Assume that cost variances are normally distributed and that both Process A and Process B cost variances are out of control. Find the probability that a cost variance for Process A will be investigated. Find the probability that a cost variance for Process B will be investigated. Which out-of-control process will be investigated more often.

Process A
Process B

(Click to select)Process BProcess A is investigated more often.

(c) If both Processes A and B are almost always in control, which process will be investigated more often.

(Click to select)Process AProcess B will be investigated more often.

(d) Suppose that we wish to reduce the probability that Process B will be investigated (when it is in control) to .2891. What cost variance investigation policy should be used? That is, how large a cost variance should trigger an investigation? (Round your final answer to the nearest whole number.)

Using this new policy, what is the probability that an out-of-control cost variance for Process B will be investigated? (Round your final answer to four decimal places.)

Cost variance
Probability that an out-of-control cost variance for Process B will be investigated

Answer 1

First, let's frame the problem from a statistical viewpoint.

There are two processes A and B for a random variable cost variance. The two processes are described using the given means and variances for the cases when the cost variances are in control and out of control. Let us denote the parameters (mean and standard deviation) for the different conditions for A and B as described below -:

Condition	Mean	Standard Deviation
Process A in control
Process A out of control
Process B in control
Process B out of control

Now, since it is given that we can assume cost variances to be distributed normally -:

(a) The probability that process A (when in control) is investigated, is essentially the probability of a random variable where   This probability can be evaluated using the standard normal CDF table , where Z is a gaussian random variable with 0 mean and a standard deviation of 1. Since when A is in control,   

One can look up the values in the standard normal CDF table to find  , which turns out to be 0.2912. To find the probabilities using standard normal CDF tables, use the fact that the gaussian PDF is symmetric and that the total area under the curve is 1. Hence, if the table gives an area from 0 to x (let that area be , then . Using the similar steps, the probability that the process B (when in control) is investigated is :

Hence, as the probability of process B being investigated ( 0.3897) is greater than process A being investigated (0.2912) when both the processes are in control, we can say that process B will be investigated more often.

(b) now since the processes are out of control, we use the same approach as in part (a), with the means and standard deviations corresponding to the out of control case for processes A and B. Therefore,

Hence Probability of Process A being investigated in out of control situation is 0.7995. Similarly, for process B, we get

is a random variable analogous to , i.e   for the controlled case.

Hence for this case, since the probability of process A being investigated is higher, it will be investigated more often as compared to process B.

(c) Already answered in part (a), that process B will be more likely to get investigated.

(d) Let the process B is now investigated if cost variance is greater than . Using the expressions above, we can write that

Using the fact that , we get that . Using the inverse table we get that x is closest to 0.56 in the look-up table. Hence,

or in other words, , which is almost 5751 dollars. Hence the probability of an out of control process B investigated with this new policy is :

Note that these answers are slightly approximate due to the least count of the standard normal table.