Question

An appliance dealer sells three different models of freezers having \(13.5,15.9,\) and 19.1 cubic feet of storage space. Let \(x=\) the amount of storage space purchased by the next customer to buy a freezer. Suppose that \(x\) has the following probability distribution: $$ \begin{array}{lrrr} x & 13.5 & 15.9 & 19.1 \\ p(x) & 0.2 & 0.5 & 0.3 \end{array} $$ a. Calculate the mean and standard deviation of \(x\). (Hint: See Example 6.15\()\) b. Give an interpretation of the mean and standard deviation of \(x\) in the context of observing the outcomes of many purchases.

Short answer

The mean value indicates the average storage space a customer is likely to purchase, while the standard deviation shows the average variation of storage space sizes chosen by customers.

Step-by-step solution

Calculating the Mean

The formula to calculate the mean or expected value 'μ' for a discrete probability distribution is: \[μ = ∑[x*p(x)]\] where 'x' represents the value of the outcome and 'p(x)' is the probability of that outcome. In this exercise, this translates to: \[μ = (13.5*0.2) + (15.9*0.5) + (19.1*0.3)\]

Calculating the Standard Deviation

The standard deviation 'σ' for a discrete probability distribution is calculated using the formula: \[σ = √{∑[(x - μ)^2 * p(x)]}\] Firstly we need to find \[(x - μ)^2 * p(x)\] for each x and then sum them up. Finally, we take the square root of this sum to get σ.

Interpreting the Mean and Standard Deviation

The mean gives the 'expected value', which can be understood as the average storage space a customer is likely to purchase. The standard deviation provides information about the 'dispersion' of the data, i.e., how much the customer's choice deviates from the mean choice on average. The larger the standard deviation, the wider the variation in customer choice.

Key concepts

Mean or Expected Value

Understanding the mean or expected value of a discrete probability distribution is fundamental in predicting outcomes in a probabilistic setting. In the context of our appliance dealer example, we want to find out what the average storage space the next customer is likely to purchase. We use the formula \[μ = ∑[x ∗ p(x)]\] where every storage space option, labeled as 'x', is multiplied by its corresponding probability 'p(x)', and then the products are summed up.
In simple terms, you can think of the mean as the 'center' of the distribution—the balance point where the probabilities are weighed by their outcomes. For the dealer, knowing the mean storage space means that over many sales, the average freezer sold would be close to this size—not too small, not too large, but just typical according to the given probabilities.
As for the calculation, the mean would be: \[μ = (13.5 ∗ 0.2) + (15.9 ∗ 0.5) + (19.1 ∗ 0.3)\] After the multiplication and addition, the mean storage space amounts to: \[μ = 2.7 + 7.95 + 5.73 = 16.38\] cubic feet, indicating this is the expected size freezer the next customer might buy.
It's also key to note that while calculating the mean helps us with long-term expectations, it doesn't always reflect the most probable individual event unless the distribution is symmetrical or has one peak.

Standard Deviation

The standard deviation is a measure of how spread out numbers are in a data set or—in our case—a probability distribution. It gives us an idea of how much the actual values can vary from the mean.
To calculate it, we first need to work out the difference between each possible outcome and the mean, square this difference, multiply it by the probability for that outcome, and then add up these weighted squared differences. The formula looks like this: \[σ = √{∑[(x - μ)^2 ∗ p(x)]}\] In practice, we are squaring the differences to avoid negative values and giving more weight to larger deviations. After computing these for each storage size option, we sum them up and finally take the square root to achieve the standard deviation.
In our freezer storage space scenario, this calculation allows the dealer to see not just the typical size freezer sold (the mean) but also the variability in the sizes that customers are purchasing. It's the diversity of customer preferences encapsulated in one metric. A larger standard deviation means a wider range of preferences, whereas a smaller one indicates that most customers are consistently buying freezers of similar sizes.

Probability Distribution Interpretation

The interpretation of a discrete probability distribution goes beyond calculating the mean and standard deviation; it involves understanding what the distribution tells us about the real-world situation it represents.
For instance, the distribution given for the appliance dealer's freezer sizes—13.5, 15.9, and 19.1 cubic feet—tells us which sizes are more or less likely to be bought. The mean, in this case, helps us anticipate that a typical customer is likely to buy a freezer around 16.38 cubic feet in size. But to gain a fuller picture, we look at the shape and spread of the distribution, which the standard deviation helps quantify.
A smaller standard deviation would indicate that most customers choose sizes close to the mean, reflecting a consistency in preference. A larger standard deviation, on the other hand, would suggest a more diverse set of choices among customers with no single freezer size dominating the sales. This nuanced understanding helps businesses to plan inventory, predict sales, and tailor marketing efforts based on the statistical behavior of their customers. Ultimately, the probability distribution serves as a model for decision-making and forecasting in uncertain conditions, which is quintessential in the business realm.