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An appliance manufacturer offers extended warranties on its washers and dryers. Based on past sales, the manufacturer reports that of customers buying both a washer and a dryer, \(52 \%\) purchase the extended warranty for the washer, \(47 \%\) purchase the extended warranty for the dryer, and \(59 \%\) purchase at least one of the two extended warranties. a. Use the given probability information to set up a "hypothetical 1000 " table. b. Use the table from Part (a) to find the following probabilities: i. the probability that a randomly selected customer who buys a washer and a dryer purchases an extended warranty for both the washer and the dryer. ii. the probability that a randomly selected customer purchases an extended warranty for neither the washer nor the dryer.

Step by step solution

01

Setting up the hypothetical table

Let's assume there are 1000 customers. This gives us a total of 520 (52% of 1000) customers will purchase the extended warranty for the washer, 470 (47% of 1000) customers will purchase the extended warranty for the dryer, and 590 (59% of 1000) customers will purchase at least one of the two extended warranties. So the number of customers who buy the warranty for both appliances would be the sum of those who buy the washer and dryer warranty minus those who buy at least one, which gives us \(520 + 470 - 590 = 400\).

02

Find the probability for buying both warranties

To find the probability a customer buys both warranties, we divide the number of customers who buy both by the total number of customers. Therefore the probability equals \(400/1000 = 0.4\) or 40%.

03

Find the probability for buying neither warranty

To find the probability a customer buys no warranty, we can subtract those who bought at least one from the total customers. Therefore, the number of customers who buy neither warranty equals \(1000 - 590 = 410\). Then divide by the total number of customers, the probability thus equal to \(410/1000 = 0.41\) or 41%.

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Key Concepts

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

Extended Warranty

Extended warranties are additional protection plans that consumers can purchase to cover repairs or replacements beyond the standard manufacturer's warranty. These plans can often save money if appliances break down unexpectedly after the manufacturer's warranty expires. In our exercise, the focus is on whether customers who purchase washers and dryers also buy extended warranties for their appliances. The extended warranty upsell is an example of analyzing consumer behavior using probability to anticipate future sales trends.

Hypothetical Table

A hypothetical table is a statistical tool used to simplify complex probability calculations by assuming a sample size, often 1000 units, to represent customer behavior. This approach was utilized in the exercise to make calculations clearer and more manageable. Using a hypothetical 1000 customers, we calculated specific data points such as:

• 520 customers purchasing the warranty for the washer
• 470 customers purchasing the warranty for the dryer
• 590 customers purchasing at least one warranty

By using these hypothetical numbers, we can find the overlap and distinctions in customer choices, leading to an easier understanding of the probability problems presented.

Statistical Problem Solving

Problem-solving in statistics often involves translating real-world scenarios into numerical and probabilistic contexts. In this exercise, statistical problem-solving allows us to determine the likelihood of different warranty purchase behaviors among customers. The steps included setting up a hypothetical table to organize our data and then performing calculations to find probabilities, like those buying both warranties or none at all. This method showcases the importance of structured data organization and logical reasoning in statistical analysis. Understanding the basics of creating hypothetical scenarios can significantly improve your problem-solving capabilities in statistics.