/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 The budgeting process for a midw... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 55

The budgeting process for a midwestern college resulted in expense forecasts for the coming year (in \(\$$ millions) of \)\$ 9, \$ 10, \$ 11, \$ 12,\( and \)\$ 13 .\( Because the actual expenses are unknown, the following respective probabilities are assigned: \).3, .2, .25, .05,\( and \).2 .\( a. Show the probability distribution for the expense forecast. b. What is the expected value of the expense forecast for the coming year? c. What is the variance of the expense forecast for the coming year? d. If income projections for the year are estimated at \)\$ 12$ million, comment on the financial position of the college.

Short Answer

Expert verified
The probability distribution shows expenses from \$9 to \$13 million with respective probabilities. The expected expense is \$10.65 million, variance 2.1275; with \$12 million income, the college is likely financially stable.

Step by step solution

01

Create Probability Distribution Table

Create a table with all the given expense forecasts and their respective probabilities. This helps to organize the information needed to display the probability distribution. | Expense Forecast ( $ million) | Probability | |---------------------|-------------| | 9 | 0.3 | | 10 | 0.2 | | 11 | 0.25 | | 12 | 0.05 | | 13 | 0.2 | The table shows the probability distribution of the expense forecasts.
02

Calculate Expected Value

The expected value (mean) of the expense forecast is calculated by using the formula:\[ E(X) = \sum (x_i \cdot p_i) \]where \( x_i \) are the expense forecasts, and \( p_i \) are the probabilities.\[ E(X) = (9 \cdot 0.3) + (10 \cdot 0.2) + (11 \cdot 0.25) + (12 \cdot 0.05) + (13 \cdot 0.2) \]\[ E(X) = 2.7 + 2 + 2.75 + 0.6 + 2.6 = 10.65 \]Thus, the expected value is \$10.65 million.
03

Calculate Variance

The variance of the expense forecast is calculated with the formula:\[ \text{Var}(X) = \sum (x_i^2 \cdot p_i) - E(X)^2 \]First, calculate \( \sum (x_i^2 \cdot p_i) \):\[ 9^2 \cdot 0.3 = 24.3 \]\[ 10^2 \cdot 0.2 = 20 \]\[ 11^2 \cdot 0.25 = 30.25 \]\[ 12^2 \cdot 0.05 = 7.2 \]\[ 13^2 \cdot 0.2 = 33.8 \]Sum these values: \[ 24.3 + 20 + 30.25 + 7.2 + 33.8 = 115.55 \]Now compute the variance:\[ \text{Var}(X) = 115.55 - (10.65)^2 \]\[ \text{Var}(X) = 115.55 - 113.4225 = 2.1275 \]The variance of the expense forecast is 2.1275.
04

Financial Position Comment

The expected expense is \\(10.65 million, and the variance is 2.1275, indicating some fluctuation. With an income projection of \\)12 million, the college is expected to cover its expenses because the income exceeds the expected expenses by \$1.35 million, providing a cushion for potential expense variations.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Expected Value
The concept of expected value is crucial in probability distribution, especially in forecasting future financial scenarios. It gives us a single number that sums up a probability distribution by representing the mean of the distribution.
To calculate the expected value, we multiply each possible outcome by its respective probability and sum up all these products.
This is represented in the formula: \[ E(X) = \sum (x_i \cdot p_i) \]
In our problem, the expected value of expense forecasts is calculated as follows:
  • Multiply each expense forecast (9, 10, 11, 12, 13) by its probability (0.3, 0.2, 0.25, 0.05, 0.2).
  • Add these products together: \[ 2.7 + 2 + 2.75 + 0.6 + 2.6 = 10.65 \]
This gives us an expected value of $10.65 million, suggesting the average expected expense of the college over many probable scenarios.
Calculating Variance in Expenses
Variance measures how much the expenses are expected to fluctuate around the expected value. A higher variance means more unpredictability, which can impact financial planning.
The variance is calculated with the formula:\[ \text{Var}(X) = \sum (x_i^2 \cdot p_i) - E(X)^2 \]
To break it down:
  • First, compute the square of each expense forecast (for example, 9 squared is 81) and multiply by their probabilities.
  • Sum these values to get \( \sum (x_i^2 \cdot p_i) \): \[24.3 + 20 + 30.25 + 7.2 + 33.8 = 115.55\]
  • Subtract the square of the expected value: \[ 115.55 - (10.65)^2 = 2.1275 \]
This results in a variance of 2.1275, indicating there is some variability in the expenses, which, although not excessively high, should be factored into the budgeting decisions.
Role of Budgeting Process in Financial Planning
Budgeting is an essential part of financial management, especially for institutions like colleges.
It involves predicting future expenses and incomes to ensure the organization can meet its goals without financial hiccup.
In this scenario, the probability distribution helps to assess different expense forecasts with associated probabilities, providing a structured approach to handling uncertainty.
With the expected expense being $10.65 million and a variance indicating potential fluctuations, the college must understand its capacity to cover these expenses with the projected income of $12 million.
This leaves a comfortable margin even after considering the variance.
Effective budgeting involves using these insights:
  • Create realistic forecasts of expenses.
  • Understand potential fluctuations using variance.
  • Plan for unforeseen circumstances by considering the worst-case scenarios.
Thus, the college can make informed decisions, knowing that their income projection exceeds the expected expenses, providing a safety net for any unexpected financial variations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Many companies use a quality control technique called acceptance sampling to monitor incoming shipments of parts, raw materials, and so on. In the electronics industry, component parts are commonly shipped from suppliers in large lots. Inspection of a sample of \(n\) components can be viewed as the \(n\) trials of a binomial experiment. The outcome for each component tested (trial) will be that the component is classified as good or defective. Reynolds Electronics accepts a lot from a particular supplier if the defective components in the lot do not exceed \(1 \% .\) Suppose a random sample of five items from a recent shipment is tested. a. Assume that \(1 \%\) of the shipment is defective. Compute the probability that no items in the sample are defective. b. Assume that \(1 \%\) of the shipment is defective. Compute the probability that exactly one item in the sample is defective. c. What is the probability of observing one or more defective items in the sample if \(1 \%\) of the shipment is defective? d. Would you feel comfortable accepting the shipment if one item was found to be defective? Why or why not?

A shipment of 10 items has two defective and eight nondefective items. In the inspection of the shipment, a sample of items will be selected and tested. If a defective item is found, the shipment of 10 items will be rejected. a. If a sample of three items is selected, what is the probability that the shipment will be rejected? b. If a sample of four items is selected, what is the probability that the shipment will be rejected? c. If a sample of five items is selected, what is the probability that the shipment will be rejected? d. If management would like a .90 probability of rejecting a shipment with two defective and eight nondefective items, how large a sample would you recommend?

The nation's two biggest cable providers are Comcast Cable Communications, with 21.5 million subscribers, and Time Warner Cable, with 11.0 million subscribers (The New York Times 2007 Almanac). Suppose that management of Time Warner Cable subjectively assessed a probability distribution for \(x\), the number of new subscribers they will obtain over the next year in the state of New York, as follows: $$\begin{array}{cc} \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\ 100,000 & .10 \\ 200,000 & .20 \\ 300,000 & .25 \\ 400,000 & .30 \\ 500,000 & .10 \\ 600,000 & .05 \end{array}$$ a. Is this probability distribution valid? Explain. b. What is the probability Time Warner will obtain more than 400,000 new subscribers? c. What is the probability Time Warner will obtain fewer than 200,000 new subscribers?

Airline passengers arrive randomly and independently at the passenger- screening facility at a major international airport. The mean arrival rate is 10 passengers per minute. a. Compute the probability of no arrivals in a one-minute period. b. Compute the probability that three or fewer passengers arrive in a one- minute period. c. Compute the probability of no arrivals in a 15 -second period. d. Compute the probability of at least one arrival in a 15 -second period.

Military radar and missile detection systems are designed to warn a country of an enemy attack. A reliability question is whether a detection system will be able to identify an attack and issue a warning. Assume that a particular detection system has a .90 probability of detecting a missile attack. Use the binomial probability distribution to answer the following questions. a. What is the probability that a single detection system will detect an attack? b. If two detection systems are installed in the same area and operate independently, what is the probability that at least one of the systems will detect the attack? c. If three systems are installed, what is the probability that at least one of the systems will detect the attack? d. Would you recommend that multiple detection systems be used? Explain.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.