91Ó°ÊÓ

Some companies, such as DemandTec, have developed software to help retail chains set prices that optimize their profits. An Associated Press story (April 28,2007 ) about this software described a case in which a retail chain sold three similar power drills: one for \(\$ 90\), a better one for \(\$ 120\), and a top-tier one for \(\$ 130\). Software predicted that by selling the middle-priced drill for only \(\$ 110,\) the cheaper drill would seem less a bargain and more people would buy the middle-price drill. a. For the original pricing, suppose \(50 \%\) of sales were for the \(\$ 90\) drill, \(20 \%\) for the \(\$ 120\) drill, and \(30 \%\) for the \(\$ 130\) drill. Construct the probability distribution of \(X=\) selling price for the sale of a drill, and find its mean and interpret, b. For the new pricing, suppose \(30 \%\) of sales were for the \(\$ 90\) drill, \(40 \%\) for the \(\$ 110\) drill, and \(30 \%\) for the \(\$ 130\) drill. Is the mean of the probability distribution of selling price higher with this new pricing strategy? Explain.

Step by step solution

01

Define Variables and Original Probability Distribution

The selling price variable, \(X\), takes values \(90\), \(120\), and \(130\), with probabilities \(P(X=90) = 0.50\), \(P(X=120) = 0.20\), and \(P(X=130) = 0.30\). This is derived from the original sales proportions.

02

Calculate Mean for Original Pricing

Compute the expected value (mean) of \(X\) for the original distribution: \[ E(X_{original}) = (90 \times 0.50) + (120 \times 0.20) + (130 \times 0.30) \] Calculate this to find: \[ E(X_{original}) = 45 + 24 + 39 = 108 \] Thus, the mean selling price for the original distribution is \$108.

03

Define New Probability Distribution

For the new distribution, the selling price takes values \(90\), \(110\), and \(130\) with probabilities \(P(X=90) = 0.30\), \(P(X=110) = 0.40\), and \(P(X=130) = 0.30\).

04

Calculate Mean for New Pricing Strategy

Compute the expected value (mean) of \(X\) for the new distribution: \[ E(X_{new}) = (90 \times 0.30) + (110 \times 0.40) + (130 \times 0.30) \] Calculate this to find: \[ E(X_{new}) = 27 + 44 + 39 = 110 \] Hence, the mean selling price for the new distribution is \$110.

05

Compare Means and Interpret

Compare the means: \\(108 (original) vs. \\)110 (new). The mean selling price has increased by \$2 with the new pricing strategy. This suggests that the new pricing strategy leads to a higher average revenue per drill sold.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value

Expected value is a crucial concept in probability theory that gives us a measure of the center of a probability distribution. In simple terms, it represents the average outcome if we repeated a random process multiple times. For example, consider the scenario of selling different models of power drills. Each model sells at a specific price with a known probability, which forms our probability distribution. To calculate the expected value of selling price for the drills, we multiply each possible price by its probability and sum these products. This calculation enables businesses to anticipate average sales revenue. Specifically, for the original pricing strategy, we computed the expected value as:- - \(= 90 \times 0.50\)- - \(= 120 \times 0.20\)- - \(= 130 \times 0.30\)Adding these gives us an expected value of \(\$108\). Such insights help retailers make informed decisions about their pricing strategies.

Pricing Strategy

A well-crafted pricing strategy can significantly affect a company's revenue. In the case of the power drills, the objective was to adjust prices to influence sales volume and increase average revenue. For instance, by lowering the price of the mid-tier drill from \(\\(120\) to \(\\)110\), customers perceived the lower-priced drill as less of a bargain, resulting in increased sales of the middle option.The change in pricing not only altered the purchase behavior but also the probability distribution of sales prices. The new distribution showed that the mid-tier drill sold more, shifting sales away from the lowest-priced drill. This shift was intentional and based on understanding consumer psychology—how price differences influence buyers' perceptions and choices.By strategically adjusting prices, businesses can optimize their product positioning in the market. They balance between competing on price and maximizing revenue.

Mean Increase

With changes in the probability distribution due to a better pricing strategy, we expect to see differences in the mean — the average selling price per drill. The mean increase reflects the effectiveness of this strategy in enhancing revenue.In our exercise, altering the price of the mid-tier drill led to a new mean selling price of \(\\(110\), up from the previous \(\\)108\). By calculating the mean in both scenarios:- - Original mean: \(E(X_{original}) = \\(108\)- - New mean: \(E(X_{new}) = \\)110\)The increase of \(\$2\) in the mean selling price indicates a successful strategy shift by boosting overall revenue. Improving the mean selling price by understanding probability distributions validates the change, encouraging further strategic decisions based on robust calculations.