Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The customer service department of Universal Instruments, manufacturer of the Galaxy home computer, conducted a survey among customers who had returned their purchase registration cards. Purchasers of its deluxe model home computer were asked to report the length of time \((t)\) in days before service was required. a. Describe a sample space corresponding to this survey. b. Describe the event \(E\) that a home computer required service before a period of 90 days had elapsed. c. Describe the event \(F\) that a home computer did not require service before a period of 1 yr had elapsed.

Step-by-Step Solution

Answer

The sample space for this survey is defined as \(S = \{t : t \in \mathbb{N}, 0 \leq t \leq 365 \}\), representing all possible lengths of time \((t)\) in days before service was required within the first year of purchase. Event \(E\) is defined as \(E = \{t \in S : t \leq 90 \}\), representing the home computers that required service within 90 days. Event \(F\) is an empty set (\(F = \emptyset\)), as our sample space only considers days within the first year, and event \(F\) pertains to home computers that did not require service before a period of 1 year had elapsed.

Step 1: Define the sample space

The sample space corresponds to the set of all possible outcomes of this survey. In this case, the possible outcomes are the number of days \((t)\) before service was required for each Galaxy home computer. We can define the sample space as: \[ S = \{t : t \in \mathbb{N}, 0 \leq t \leq 365 \} \] where \(\mathbb{N}\) denotes the set of natural numbers. Note that we have limited the maximum number of days to 365 because we're interested in the service required within the first year of the purchase.

Step 2: Describe event E

Event \(E\) is the event that a home computer required service before a period of 90 days had elapsed. In terms of our sample space, \(E\) can be described as the set of outcomes where the length of time \((t)\) before service required is less than or equal to 90 days: \[ E = \{t \in S : t \leq 90 \} \]

Step 3: Describe event F

Event \(F\) is the event that a home computer did not require service before a period of 1 year (365 days) had elapsed. In terms of our sample space, \(F\) can be described as the set of outcomes where the length of time \((t)\) before service required is greater than 365 days: \[ F = \{t \in S : t > 365 \} \] However, since we have defined our sample space \(S\) as only containing the days within the first year, the event \(F\) is, in fact, an empty set: \[ F = \emptyset \] This means that according to our sample space, there is no home computer that did not require service before a period of 1 year had elapsed.

Key Concepts

Event Probability

When exploring the concept of event probability, we look into the likelihood of an event occurring within a given sample space. For students to grasp this concept, it's essential to understand that the sample space consists of all possible outcomes of a particular experiment or survey. In the example of the Galaxy home computer survey, we define event probability as the fraction of outcomes in the sample space that comprise the event in question. Understanding event probability is fundamental for analyzing survey data and making informed decisions based on statistical findings.

Improving Event Probability Analysis

One way to make this concept clearer is to discuss the relative frequency approach. It helps to think of the probability of an event as the number of times the event occurs divided by the total number of trials, as long as the number of trials is large enough. For example, if a sizeable number of customers reported needing service within the first 90 days, that probability could be approximated as:

\[ P(E) = \frac{\text{Number of computers needing service within 90 days}}{\text{Total number of surveyed computers}} \.\]

Survey Statistical Analysis

Survey statistical analysis includes collecting data, summarizing it, and making inferences about a larger population based on the sample collected. In the context of customer surveys for the Galaxy home computer, statistical analysis might involve calculating measures of central tendency or dispersion of the time until service is needed, or estimating the probability of certain events concerning service requirements.

Interpreting Survey Data

For a more in-depth understanding, one should also consider the principles of designing surveys and collecting data to ensure that the analysis yields meaningful and unbiased results. This might involve discussing random sampling, which helps ensure that the sample is representative of the overall customer population. In the discussion of Galaxy computers, one might consider whether customers who return registration cards represent all purchasers accurately, as this could affect the survey's outcomes.

• Understanding Sample Representativeness
• Assessing Tools for Data Collection
• Applying Proper Analysis Techniques

Set Theory in Mathematics

Set theory is a foundational discipline in mathematics that deals with the study of sets or collections of objects and their properties. A set is defined by its elements, and in the example of the Galaxy home computer survey, the sample space and events can be regarded as sets with certain characteristics. We defined the sample space (S) as the set of all natural numbers representing the days until service is needed, and the events (E and F) as subsets of this space with certain conditions applied.

Visualizing Sets and Operations

One practical approach to visualize set theory concepts is through Venn diagrams, which can illustrate how different sets (events) overlap or differ. For instance, the event E—computers needing service within 90 days—is a subset of S visualized as a circle within a larger circle representing S. To clarify the concept of an empty set (F), a separate, non-intersecting circle might be shown to indicate that there are no elements within one year that meet the criteria for the set F, corresponding to our problem statement.