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Chapter 13: Problem 51

A company's records indicate that on any given day about \(1 \%\) of their day- shift employees and \(2 \%\) of the night-shift employees will miss work. Sixty percent of the employees work the day shift. a. Is absenteeism independent of shift worked? Explain. b. What percent of employees are absent on any given day?

Short Answer

Expert verified
a) No, absenteeism is not independent of the shift worked because the probability of an employee being absent depends on whether they work the daytime or nighttime shift. b) The percentage of employees that are absent on any given day could be calculated using the weighted average formula described in step 2.

Step by step solution

01

Determine if Absenteeism and Shift worked are Independent

A pair of events is independent if the occurrence of one event doesn't affect the occurrence of the second event. The formula used to determine this is: P(A 鈭� B) = P(A)P(B) where A and B are two events.Translating that into the context of this problem: The event A is that an employee is absent. The event B is the employee working factor, divided into day shift (B1) and night shift (B2).We are given: P(A|B1) = 0.01 (probability of an employee being absent given they work on the day shift) and P(A|B2) = 0.02 (probability of an employee being absent given they work on the night shift). Also, we are given that P(B1) = 0.6 (the probability of an employee working the day shift) and therefore P(B2) = 0.4 because the probability of all possible outcomes (either working the day or night shift in this case) must sum up to 1. If P(A|B1) = P(A|B2), that would indicate that whether someone works the day shift or the night shift does not affect the likelihood of them being absent, meaning the two events are independent. However, since P(A|B1) 鈮� P(A|B2), these events cannot be independent, because the probability of absence depends on whether the employee works on the day or night shift.
02

Calculate percent of employees that are absent on any given day

The overall probability of an employee being absent is the weighted average of the probabilities of absence in each shift. It's computed as follows: P(A) = P(A and B1) + P(A and B2) = P(A|B1)P(B1) + P(A|B2)P(B2) = 0.01(0.6) + 0.02(0.4)This step gives us an understanding of the total amount of employees that miss work any given day.

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Key Concepts

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

Independent Events
In statistics, understanding whether events are independent is crucial for accurate analysis. Two events, A and B, are considered independent if the occurrence of one does not affect the probability of the other happening. This means the condition: \( P(A \cap B) = P(A) \cdot P(B) \) must hold true for events A and B to be independent.

When applied to the context of workplace absenteeism, let's consider event A as an employee being absent, and event B as the shift they work, either day or night. To check for independence, we compare the probabilities \( P(A|B_1) \) and \( P(A|B_2) \), where \( B_1 \) is the day shift and \( B_2 \) is the night shift.
  • The given probabilities are: \( P(A|B_1) = 0.01 \) and \( P(A|B_2) = 0.02 \).
  • If \( P(A|B_1) = P(A|B_2) \), then absenteeism does not depend on the shift.
  • Since \( P(A|B_1) eq P(A|B_2) \), absenteeism is indeed dependent on the shift worked.
Thus, absenteeism is influenced by whether employees are scheduled for day or night shifts, indicating a lack of independence between these two events.
Conditional Probability
Conditional probability helps us find the likelihood of an event occurring given another specific event has taken place. This is expressed as \( P(A|B) \), the probability of event A happening given that event B is true.

In our example, we examine how absenteeism (event A) varies based on whether an employee works the day shift (event \( B_1 \)) or the night shift (event \( B_2 \)). The goal is to understand how the condition of working particular shifts influences the likelihood of being absent.
  • We know that \( P(A|B_1) = 0.01 \), meaning there's a 1% chance of absence if the employee works during the day.
  • Similarly, \( P(A|B_2) = 0.02 \), indicating a 2% chance if they work the night shift.
This analysis lets us see how specific conditions, like shift timing, affect other outcomes, such as absenteeism. Thus, conditional probability is a powerful tool for predicting the likelihood of an event based on another event.
Weighted Average
A weighted average is used when different categories have varying degrees of importance or relevance. It allows us to find a combined average proportionally, taking the weights into account.

Let's apply this to find the overall proportion of employees absent. In this scenario, the weights are the probabilities of employees working either the day shift or the night shift:
  • Day shift probability: \( P(B_1) = 0.6 \)
  • Night shift probability: \( P(B_2) = 0.4 \)
To calculate the total percentage of absentees, the following weighted average formula is applied: \[ P(A) = P(A|B_1) \cdot P(B_1) + P(A|B_2) \cdot P(B_2) = 0.01 \times 0.6 + 0.02 \times 0.4\]

This calculation provides us with the overall absenteeism rate, considering both shifts together based on their occurrence probability. Using weighted averages helps to give a more accurate picture when dealing with different groups or conditions.

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Most popular questions from this chapter

A junk box in your room contains a dozen old batteries, five of which are totally dead. You start picking batteries one at a time and testing them. Find the probability of each outcome. a. The first two you choose are both good. b. At least one of the first three works. c. The first four you pick all work. d. You have to pick five batteries to find one that works.

PewInternet reported in January of 2014 that \(32 \%\) of U.S. adults own at least one e-reader, and that \(28 \%\) of U.S. adults read at least one e-book in the previous year (and thus, presumably, owned an ereader). Overall, \(76 \%\) of U.S. adults read at least one book (electronic or otherwise) in the previous year. a. Explain how these statistics indicate that owning an e-reader and reading at least one book in the previous year are not independent. b. What's the probability that a randomly selected U.S. adult has an ereader but didn't use it to read an e-book in the previous year?

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