Should We Purchase Extended Warranties For Household Product ✓ Solved

1 Should we Purchase Extended Warranties for Household Products?

Extended warranties for household products, particularly electronics, are offers that extend the coverage period beyond the manufacturer's warranty, typically by 2-3 years. These warranties generally cost between 10-20% of the product's purchase price. The decision to purchase an extended warranty depends on evaluating the probability of product failure and the associated costs of repair or replacement.

From a probability theory perspective, assessing whether to buy an extended warranty involves estimating the likelihood that the product will fail within the warranty period. If the probability of failure is high during this extended period, purchasing the warranty becomes more justifiable, as it can mitigate significant unexpected costs. Conversely, if the product is statistically likely to last beyond the warranty period, the purchase may not be financially beneficial.

For example, suppose the failure probability of a household appliance within three years is 0.2 (20%), and the cost of repair/replacement is $1,000. The expected cost without a warranty is $200 (0.2 x $1,000). If the extended warranty costs $200 (20% of a $1,000 item), purchasing it could be justified if the actual failure probability exceeds 0.2. For low-probability failures, the warranty might not be cost-effective, and saving money could be preferable.

Therefore, it is advisable to purchase extended warranties selectively rather than for all household products. Items with higher failure probabilities or more expensive repair costs merit protection, whereas low-cost, durable items may not warrant the additional expense.

2 Meaningful Sampling Methods

To determine the popularity of a bill in a city of 500,000 residents, an efficient and unbiased sampling method must be employed. A suitable approach is stratified random sampling, where the population is divided into different strata, such as age groups, neighborhoods, or socioeconomic levels. Random samples are then taken from each stratum proportionally to their representation in the population. This method ensures diverse representation and reduces bias.

The sample size should be sufficient to achieve a desired confidence level and margin of error. For example, to estimate the bill’s popularity within ±3% at a 95% confidence level, a sample size of approximately 385 individuals might suffice, based on standard sample size calculations for proportions. This formula considers the population size, proportion estimate, confidence level, and margin of error.

Sending out surveys via mail can be less costly but might introduce bias if certain demographics are less likely to respond. Meanwhile, door-to-door surveys might be more resource-intensive but can achieve higher response rates and better representativeness. A mixed approach, combining mail and personal interviews, can be an effective way to maximize coverage and minimize bias.

The parameter of interest is the proportion of the population supporting the bill, denoted as p. The sampling error is the margin of error, which quantifies the range within which the true population proportion likely falls, given the sample estimate.

3 Choosing the Right Measure of Central Tendency

The measure of central tendency describes the typical or average value in a data set:

• Mode: The most frequently occurring value in a data set. It is useful for categorical data or identifying the most common category or response.
• Mean: The arithmetic average calculated by summing all values and dividing by the number of values. The mean is best used when data are symmetrically distributed without outliers.
• Median: The middle value when the data are ordered from smallest to largest. The median is most appropriate for skewed distributions or when outliers are present, as it is resistant to extreme values.
• Mid-range: The average of the minimum and maximum values in the data set. It provides a quick estimate of the center but is sensitive to outliers and less commonly used.

Examples:

• Mode: To find the most common favorite ice cream flavor among children, which might be chocolate.
• Mean: Calculating the average test score of a class where scores are evenly distributed.
• Median: Determining the typical house price in a neighborhood with some extremely expensive properties, where the median provides a better central value than the mean.
• Mid-range: Estimating the typical temperature range in a city by averaging the lowest and highest recorded temperatures over a year.

4 Revisiting our Beginnings

Reflecting on the journey through this course, my understanding of finance and statistics has significantly deepened. Initially, concepts like probability, averages, and sampling seemed abstract, but over time, I learned to apply these ideas practically. The coursework facilitated a clearer grasp of how statistical principles underpin financial decision-making and data interpretation.

I now see how statistical literacy is essential for evaluating data critically, whether analyzing investment risks, understanding market trends, or making informed consumer choices. The ability to assess probabilities helps me better evaluate potential outcomes and make decisions rooted in evidence rather than intuition.

Professionally, this knowledge enhances my capacity to interpret financial reports, assess risks, and communicate complex data insights effectively. Personally, I am more conscious of how data influences everyday decisions, such as purchasing insurance or evaluating household investments.

In the future, I plan to continue engaging with statistical data through courses, professional reading, and real-world applications. Becoming proficient in statistical literacy empowers me to be a better data consumer, making informed choices based on sound analysis. The skills gained during this course will serve as a foundation for ongoing learning and responsible data use in various aspects of life and work.

References

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