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Chapter 2: Problem 7

Workers at a particular mining site receive an average of 35 days paid vacation, which is lower than the national average. The manager of this plant is under pressure from a local union to increase the amount of paid time off. However, he does not want to give more days off to the workers because that would be costly. Instead he decides he should fire 10 employees in such a way as to raise the average number of days of that are reported by his employees. In order to achieve this goal, should he fire employees who have the most number of days off, least number of days off, or those who have about the average number of days off?

Short Answer

Expert verified
Fire employees with the least number of days off.

Step by step solution

01

Define the Problem

We need to determine which 10 employees, when removed, will increase the average number of paid vacation days for the remaining employees. We have three options: removing employees with the most, least, or around the average number of days off.
02

Understand the Impact on Average

The average of a dataset is defined as the sum of all data points divided by the number of data points. Removing data points can increase the average if points below the current average are removed, as it reduces the denominator more significantly than the numerator.
03

Theory of Averages

To increase the average, we need to reduce the total sum of the dataset less than proportional to the reduction in count. Thus, we should remove the employees with fewer days off because it will remove smaller values from the total, pushing the average up.
04

Applying the Theory

By removing employees who have fewer vacation days, we effectively increase the ratio of the total sum of vacation days to the number of employees because we're removing values that pull the average down.

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Key Concepts

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

Average Calculation
The concept of average calculation is foundational in statistics. It provides a general idea of the central tendency of a dataset. The average, also known as the mean, is calculated by summing all the data points and dividing that sum by the total number of data points. For example, if you have five employees with vacation days of 20, 25, 30, 35, and 45, the average is calculated as \[\frac{20 + 25 + 30 + 35 + 45}{5} = \frac{155}{5} = 31.\] In the context of the exercise, the manager is looking at ways to increase this average number of vacation days for the employees remaining after firing some of them.
Data Manipulation
Data manipulation involves changing the data in such a way that desired outcomes, such as a higher average, are achieved. In the given scenario, removing certain employees is an act of manipulating the data. There are different ways to manipulate:
  • Removing data points
  • Weighting data differently
  • Transforming the dataset
The manager needs to decide which group of employees, when removed, would strategically increase the average vacation days for those who remain. Here, manipulation is focused on removing lower data points from the dataset to increase the average.
Theory of Averages
The theory of averages delves into how we view and interpret average values. It tells us that the sum of values and the count of values both affect the average. If you aim to increase the average, you must decrease the denominator, i.e., remove data points with values lower than the average or increase the numerator by adding larger values. In the case at hand, removing employees with fewer vacation days effectively lowers the denominator without greatly reducing the sum, since those days are less than the average. Doing so should increase the average vacation days remaining because you're effectively removing those data points that bring down the overall average.
Problem Solving in Statistics
Problem solving in statistics involves understanding, applying, and sometimes manipulating statistical concepts to achieve a specific goal. Here, the manager's problem-solving task is to increase the average paid vacation days without actually increasing the days given. This requires a strategic approach:
  • Identify the goal: Raise the average vacation days.
  • Look at options: Remove employees with the fewest vacation days.
  • Apply logic: Calculate and predict the impact of removing specific data points.
These steps help ensure that changes to the dataset achieve the desired increase in average, without needing to provide more vacation days. This illustrates how statistical reasoning can be used practically to influence outcomes in a business context.

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Most popular questions from this chapter

For each of the following, state whether you expect the distribution to be symmetric, right skewed, or left skewed. Also specify whether the mean or median would best represent a typical observation in the data, and whether the variability of observations would be best represented using the standard deviation or IQR. Explain your reasoning. (a) Housing prices in a country where \(25 \%\) of the houses cost below $$\$ 350,000,50 \%$$ of the houses cost below $$\$ 450,000,75 \%$$ of the houses cost below $$\$ 1,000,000$$ and there are a meaningful number of houses that cost more than $$\$ 6,000,000$$. (b) Housing prices in a country where \(25 \%\) of the houses cost below $$\$ 300,000,50 \%$$ of the houses cost below $$\$ 600,000,75 \%$$ of the houses cost below $$\$ 900,000$$ and very few houses that cost more than $$\$ 1,200,000 .$$ (c) Number of alcoholic drinks consumed by college students in a given week. Assume that most of these students don't drink since they are under 21 years old, and only a few drink excessively. (d) Annual salaries of the employees at a Fortune 500 company where only a few high level executives earn much higher salaries than all the other employees.

Office productivity is relatively low when the employees feel no stress about their work or job security. However, high levels of stress can also lead to reduced employee productivity. Sketch a plot to represent the relationship between stress and productivity.

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Midrange. The midrange of a distribution is defined as the average of the maximum and the minimum of that distribution. Is this statistic robust to outliers and extreme skew? Explain your reasoning.

Facebook data indicate that \(50 \%\) of Facebook users have 100 or more friends, and that the average friend count of users is \(190 .\) What do these findings suggest about the shape of the distribution of number of friends of Facebook users? \(^{17}\)
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