Question

Suppose that for a given computer salesperson, the probability distribution of \(x=\) the number of systems sold in one month is given by the following table: $$ \begin{array}{lrrrrrrrr} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ p(x) & .05 & .10 & .12 & .30 & .30 & .11 & .01 & .01 \end{array} $$ a. Find the mean value of \(x\) (the mean number of systems sold). b. Find the variance and standard deviation of \(x\). How would you interpret these values? c. What is the probability that the number of systems sold is within 1 standard deviation of its mean value? d. What is the probability that the number of systems sold is more than 2 standard deviations from the mean?

Short answer

The mean number of systems sold is 4.19, the variance is 1.5279, and the standard deviation is 1.2365. The probability of sales within one standard deviation of the mean is 0.72, and the probability of sales more than 2 standard deviations away from the mean is 0.07.

Step-by-step solution

Calculate Mean of Sales

Multiply each sales outcome by its associated probability, and sum them together: \[ \mu = (1*.05) + (2*.10) + (3*.12) + (4*.30) + (5*.30) + (6*.11) + (7*.01) + (8*.01) = 4.19 \]

Calculate Variance

To find the variance, square the difference between each sales figure and the mean, multiply by the corresponding probability, and sum up the results: \[ \sigma^2 = ((1-4.19)^2*.05) + ((2-4.19)^2*.10) + ((3-4.19)^2*.12) + ((4-4.19)^2*.30) + ((5-4.19)^2*.30) + ((6-4.19)^2*.11) + ((7-4.19)^2*.01) + ((8-4.19)^2*.01) = 1.5279 \]

Calculate Standard Deviation

The standard deviation is the square root of the variance: \[ \sigma = \sqrt{1.5279} = 1.2365 \]

Probability Within 1 Standard Deviation

The range within one standard deviation of the mean is \(4.19-1.2365 = 2.95\) and \(4.19+1.2365 = 5.43\). Count the probabilities for the outcomes in this range: \[ P(2.95 < x < 5.43) = p(3) + p(4) + p(5) = 0.12 + 0.30 + 0.30 = 0.72 \] Since numbers of sales are integer values, it makes sense to consider whole numbers within this range.

Probability More Than 2 Standard Deviations from Mean

The range more than two standard deviations from the mean is less than \(4.19-2*1.2365 = 1.72\) and greater than \(4.19+2*1.2365 = 6.66\). Count the probabilities for the outcomes in this range: \[ P(x<1.72) = p(1) = 0.05 \] and \[ P(x>6.66) = p(7) + p(8) = 0.01 + 0.01 = 0.02 \] Total probability is therefore 0.05 + 0.02 = 0.07.

Key concepts

Understanding Mean Value in Probability Distributions

The mean value of a probability distribution, also known as the expected value, is the average number of events you can expect to occur. To find the mean value, you multiply each possible outcome by its probability and then sum all these products.

In the context of our example with the computer salesperson, if you average out the number of computer systems sold over a long period, despite the natural fluctuation in monthly sales, you would expect the average to be close to 4.19 systems per month. This figure is derived from the provided probability distribution of sales outcomes and their corresponding probabilities.

Here's a simple view of the process:

• List every potential sales outcome.
• Multiply each by its likelihood of occurring (its probability).
• Add up these products to get the mean number of systems sold per month, representing what the salesperson can expect on average.

Understanding the mean gives you a baseline to consider fluctuations in monthly sales numbers.

Variance and How It Reflects Sales Predictability

Variance in probability distributions measures the spread of outcomes around the mean value. It shows us how much, on average, each outcome varies from the expected average sales, giving insight into the predictability of sales.

If the variance is high, it means that actual sales figures deviate widely from the mean, indicating unpredictability in sales. A low variance suggests that the sales figures are more consistently close to the mean value, which signifies stable sales patterns.

To compute the variance:

• Subtract the mean from each sales figure to find the 'deviation'.
• Square each deviation to make sure they are positive values.
• Multiply each squared deviation by its corresponding probability.
• Sum these values to get the variance for the sales outcomes.

In our exercise, the variance is 1.5279. This indicates that, while there is some fluctuation in monthly sales, it is fairly moderate, as reflected by a variance that is not excessively high.

Interpreting Standard Deviation in Sales Performance

The standard deviation is a key statistical tool that tells us about the dispersion of data points in a distribution, specifically, it is the square root of the variance. In practical terms, it's often more intuitive than variance because it is expressed in the same units as the mean value.

For our sales example, a smaller standard deviation would suggest that the salesperson's monthly numbers are generally close to the mean value, signaling consistency in performance. A larger standard deviation would indicate greater variability in the number of systems sold month to month.

In this exercise, the standard deviation being 1.2365 means that most of the time, the salesperson's monthly sales will be within 1.2365 units of the mean value, 4.19. To put it simply, this person's sales figures tend to fluctuate within a reasonably narrow range, evidencing good stability in their sales performance.