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Chapter 12: Problem 13

An engineering company employs three architects that are responsible for cost estimates of new projects. Antonio makes \(30 \%\) of the estimates, Richard makes \(20 \%\) and Marco \(50 \% .\) Like all estimates, there are usually some errors in these estimates. The record of percentages of'serious'errors that cost the company thousands of euros shows Antonio at \(3 \%,\) Richard at \(2 \%,\) and Marco at \(1 \% .\) Which of the three engineers is probably responsible for most of the serious errors?

Short Answer

Expert verified
Antonio is responsible for most of the serious errors.

Step by step solution

01

Determine Contribution to Serious Errors

To find out who is responsible for the most serious errors, calculate the percentage of serious errors contributed by each architect. We will multiply the percentage of total estimates by the percentage of serious errors for each architect.
02

Calculate Antonio's Contribution

Antonio makes 30% of the estimates, with a 3% error rate: \(0.30 \times 0.03 = 0.009\). This means Antonio contributes 0.9% to the serious errors.
03

Calculate Richard's Contribution

Richard makes 20% of the estimates, with a 2% error rate: \(0.20 \times 0.02 = 0.004\). This means Richard contributes 0.4% to the serious errors.
04

Calculate Marco's Contribution

Marco makes 50% of the estimates, with a 1% error rate: \(0.50 \times 0.01 = 0.005\). This means Marco contributes 0.5% to the serious errors.
05

Compare Contributions

Compare the contributions: Antonio contributes 0.9%, Richard 0.4%, and Marco 0.5% toward the serious errors. Therefore, Antonio is responsible for the most serious errors.

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Key Concepts

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

Error Analysis
Every task we perform can have some level of error, and in business, these errors can have financial consequences. Error analysis involves identifying, quantifying, and analyzing errors in a process. For the engineering company, this process translates into understanding which architect's estimates contribute most to serious errors.
By breaking down the percentage of serious errors each architect contributes to, the company can evaluate how much of the total error each person's estimates create.
This analysis helps in targeting improvements and understanding risks. By calculating the error rate and the contribution of each designer, it is possible to prioritize solutions where they are needed most.
  • Antonio: 0.9% contribution
  • Richard: 0.4% contribution
  • Marco: 0.5% contribution
Percentages
Percentages are a way to express a number as a fraction of 100. They are used widely in probability and statistics to denote proportions and ratios, making them crucial in various analyses. In the exercise, percentages illustrate both the proportion of estimates each architect makes and their respective error rates.
Using percentages simplifies understanding and comparing data, as it standardizes the different totals for uniform comparison. To find the percentage contribution to errors:
  • Multiply the overall percentage of estimates made by each architect by their error rate.
  • This calculates what portion of all serious errors can be attributed to each engineer.
This method can be employed across various fields to measure performance, success rates, or error rates.
Mathematical Contribution
Mathematical contribution in this context refers to each architect's estimated influence on serious errors. For complex systems, knowing each component's contribution is vital for performance improvement. By multiplying the percentage of estimates by the error rate, we find how much each architect is mathematically responsible for serious errors.
  • Antonio's 0.9% shows he is leading in error contribution. This highlights areas needing focus and redesign.
  • Richard contributes a comparatively lower 0.4%, indicating efficiency or perhaps having fewer estimates.
  • Marco stands in the middle with a 0.5% contribution, still requiring vigilance for decreasing errors.
Understanding mathematical contributions helps businesses address specific areas of concern, thereby enhancing overall efficiency and effectiveness.
Statistics
Statistics provides tools and methodologies to collect, analyze, interpret, and present data systematically. In this scenario, statistics helps compare and quantify the contributions to serious errors by each architect. By determining the relative portion of errors each person contributes, statistical methods offer insights that raw numbers cannot. Statistical understanding aids in:
  • Making informed decisions by providing a clear picture of each architect's responsibility.
  • Implementing corrective actions where needed.
  • Enhancing resource allocation to minimize errors.
Through statistics, businesses can track performance and outcomes over time, refining processes based on empirical evidence rather than guesswork.

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Most popular questions from this chapter

In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate. Give reasons for your answer. a) A die is loaded such that the probability of each face is according to the following assignment ( \(x\) is the number of spots on the upper face and \(P(x)\) is its probability.) $$\begin{array}{c|cccccc} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline P(x) & 0 & \frac{1}{6} & \frac{1}{3} & \frac{1}{3} & \frac{1}{6} & 0 \\\ \hline \end{array}$$ b) A student at your school categorized in terms of gender and whether they are diploma candidates or not. P(female, diploma candidate) \(=0.57,\) P(female, not a diploma candidate) \(=0.23\) \(\mathrm{P}\) (male, diploma candidate) \(=0.43, \mathrm{P}\) (male, not a diploma candidate) \(=0.18\) c) Draw a card from a deck of 52 cards ( \(x\) is the suit of the card and \(P(x)\) is its probability). $$\begin{array}{|c|c|c|c|c|} \hline x & \text { Hearts } & \text { Spades } & \text { Diamonds } & \text { Clubs } \\ \hline P(x) & \frac{12}{52} & \frac{15}{52} & \frac{12}{52} & \frac{13}{52} \\\ \hline \end{array}$$

A construction company is bidding on three projects: \(B_{1}, B_{2}\) and \(B_{3}\). From previous experience they have the following probabilities of winning the bids: \(P\left(B_{1}\right)=0.22, P\left(B_{2}\right)=0.25\) and \(P\left(B_{3}\right)=0.28 .\) Winning the bids is not independent one from another. The joint probabilities are given below. $$\begin{array}{|c|c|c|c|} \hline & B_{1} & B_{2} & B_{3} \\ \hline B_{1} & & 0.11 & 0.05 \\ \hline B_{2} & 0.11 & & 0.07 \\ \hline B_{3} & 0.05 & 0.07 & \\ \hline \end{array}$$ Also, \(P\left(B_{1} \cap B_{2} \cap B_{3}\right)=0.01 .\) Find the following probabilities: a) \(P\left(B_{1} \cup B_{2}\right)\) b) \(P\left(B^{\prime}_{1} \cap B_{2}^{\prime}\right)\) c) \(P\left(B^{\prime}, \cap B_{2}^{\prime}\right) \cup B_{3}\) d) \(P\left(B^{\prime}_{1} \cap B_{2}^{\prime} \cap B_{3}\right)\) e) \(P\left(B_{2} \cap B_{3} | B_{1}\right)\) \(f) P\left(B_{2} \cup B_{3} | B_{1}\right)\)

A die is constructed in a way that a 1 has the chance to occur twice as often as any other number. a) Find the probability that a 5 appears. b) Find the probability an odd number will occur.

When Olympic athletes are tested for illegal drug use (doping), the results of a single test are used to ban the athlete from competition. In an experiment on 1000 athletes, 100 were using the testosterone drug. During the medical examination, the available test would positively identify \(50 \%\) of the users. It would also falsely identify \(9 \%\) of the non-users as users. If an athlete tests positive, what is the probability that he/she is really doping?

132 of 300 patients at a hospital are signed up for a special exercise program which consists of a swimming class and an aerobics class. Each of these 132 patients takes at least one of the two classes. There are 78 patients in the swimming class and 84 in the aerobics class. Find the probability that a randomly chosen patient at this hospital is a) not in the exercise program b) enrolled in both classes.
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