Problem 194 - Applied Statistics and Probability for Engineers

Question

Incoming calls to a customer service center are classified as complaints \((75 \%\) of calls) or requests for information \((25 \%\) of calls \() .\) Of the complaints, \(40 \%\) deal with computer equipment that does not respond and \(57 \%\) deal with incomplete software installation; in the remaining \(3 \%\) of complaints, the user has improperly followed the installation instructions. The requests for information are evenly divided on technical questions \((50 \%)\) and requests to purchase more products \((50 \%)\) (a) What is the probability that an incoming call to the customer service center will be from a customer who has not followed installation instructions properly? (b) Find the probability that an incoming call is a request for purchasing more products.

Step-by-Step Solution

Answer

(a) The probability is 0.0225. (b) The probability is 0.125.

Step 1: Understanding the Problem

We need to analyze incoming calls to a customer service center, which are classified into two categories with certain subcategories for complaints and requests for information.

Step 2: Calculate Probability for Improper Installation

First, we calculate the probability of calls that are complaints about improperly followed installation instructions. This is computed by multiplying the probability of a complaint with the probability of this specific type of complaint within complaints:\[ P(\text{Improper Instruction}) = P(\text{Complaint}) \times P(\text{Improper Instruction} | \text{Complaint}) = 0.75 \times 0.03 = 0.0225 \]

Step 3: Calculate Probability for Purchasing Requests

Next, we calculate the probability of calls that request to purchase more products. These requests are a subset of requests for information:\[ P(\text{Purchasing Request}) = P(\text{Request for Information}) \times P(\text{Purchasing Request} | \text{Request for Information}) = 0.25 \times 0.50 = 0.125 \]

Key Concepts

Conditional Probability

Conditional probability is a key concept in statistics that allows us to calculate the likelihood of an event occurring given that another event has already occurred. For instance, if we know a call at a customer center is a complaint, we can find the probability that this complaint involves improper installation instructions.

Conditional probability is calculated using the formula:

• \( P(A | B) = \frac{P(A \cap B)}{P(B)} \)

where \(P(A | B)\) is the probability of event \(A\) occurring given that \(B\) has happened, \(P(A \cap B)\) is the probability of both events occurring, and \(P(B)\) is the probability of event \(B\).

In our exercise, to find the probability of a call about improper installation, we see that 3% of complaints are such. Thus, given that a call is a complaint, the probability it’s due to improper installation is calculated as follows:

• \(P(\text{Improper Instruction} | \text{Complaint}) = 0.03\).

Using the total probability for complaints and applying the conditional probability, we find:

• \(P(\text{Improper Instruction}) = 0.75 \times 0.03 = 0.0225\).

This means that the probability of a random incoming call being about improper installation following the steps outlined, is 2.25%.

Probability Calculation

Probability calculation involves determining how likely an event is to occur. In the example, we needed to find probabilities for specific types of customer calls.

Let's break it down with some formulas and examples:

• General Probability Formula: \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

This means that if you want to determine the probability of a specific event, you divide the number of ways that event can occur by the total number of all possible events.

For the problem, consider the request for purchasing products:

• \(P(\text{Request for Information}) = 0.25\)
• \(P(\text{Purchase Request}| \text{Request for Information}) = 0.50\)
Calculating the probability for purchase requests, you multiply these probabilities: \[ P(\text{Purchasing Request}) = 0.25 \times 0.50 = 0.125 \].

• This indicates that there is a 12.5% chance an incoming call is for purchasing products.
Understanding these calculations helps build a general intuition for tackling various probability problems.

Categorical Data Analysis

Categorical data analysis involves organizing data into categories or groups that can be analyzed for patterns and probability. Such categories are clearly defined in our exercise and typically include identifiers like customer complaints or information requests.

Consider how the data is divided:

• Complaints:
  • Equipment Issues
  • Incomplete Software
  • Improper Instructions
• Requests for Information:
  • Technical Questions
  • Purchase Requests

By categorizing data in this way, businesses can better understand where most of their service issues lie, which helps in resource allocation and process improvements. Analyzing such data can also offer insights into customer needs and help tailor different strategies for better customer satisfaction.

Therefore, categorical data analysis not only aids in laying out clear categories but also assists in applying the right statistical methods such as probability or conditional probability to draw meaningful conclusions.