Problem 40 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a survey conducted in the fall 2006, 800 homeowners were asked about their expectations regarding the value of their home in the next few years; the results of the survey are summarized below: $$ \begin{array}{lc} \hline \text { Expectations } & \text { Homeowners } \\ \hline \text { Decrease } & 48 \\ \hline \text { Stay the same } & 152 \\ \hline \text { Increase less than } 5 \% & 232 \\ \hline \text { Increase 5-10\% } & 240 \\ \hline \text { Increase more than 10\% } & 128 \\ \hline \end{array} $$ If a homeowner in the survey is chosen at random, what is the probability that he or she expected his or her home to a. Stay the same or decrease in value in the next few years? b. Increase \(5 \%\) or more in value in the next few years?

Step-by-Step Solution

Answer

a. The probability that a randomly chosen homeowner expects their home value to stay the same or decrease is \(25\%\). b. The probability that a randomly chosen homeowner expects their home value to increase 5% or more is \(46\%\).

Step 1: Identify the number of homeowners with the desired expectations

We are given that 48 homeowners expect the value to decrease, and 152 homeowners expect the value to stay the same. Add these two numbers together to find the total number of homeowners who expect the value to either stay the same or decrease. 48 + 152 = 200

Step 2: Calculate the probability

To calculate the probability of a randomly chosen homeowner expecting their home value to stay the same or decrease, divide the number we found in Step 1 (200) by the total number of homeowners surveyed (800). Probability = \(\frac{200}{800}\)

Step 3: Simplify the fraction and convert to percentage

Simplify the fraction and convert it to a percentage: \(\frac{200}{800}\) = \(\frac{1}{4}\) = \(0.25\) = \(25\%\) So, the probability that a randomly chosen homeowner expects their home value to stay the same or decrease is \(25\%\). #b. Increase 5% or more in value in the next few years?#

Step 1: Identify the number of homeowners with the desired expectations

We are given that 240 homeowners expect the value to increase 5-10%, and 128 homeowners expect the value to increase more than 10%. Add these two numbers together to find the total number of homeowners who expect the value to increase 5% or more. 240 + 128 = 368

Step 2: Calculate the probability

To calculate the probability of a randomly chosen homeowner expecting their home value to increase 5% or more, divide the number we found in Step 1 (368) by the total number of homeowners surveyed (800). Probability = \(\frac{368}{800}\)

Step 3: Simplify the fraction and convert to percentage

Simplify the fraction and convert it to a percentage: \(\frac{368}{800}\) = \(\frac{23}{50}\) = \(0.46\) = \(46\%\) So, the probability that a randomly chosen homeowner expects their home value to increase 5% or more is \(46\%\).

Key Concepts

Survey Data Analysis

Survey data analysis involves collecting and interpreting data from a group of respondents to uncover patterns, preferences, or predictions. This process is crucial in deriving meaningful insights from raw data. In the context of the homeowner survey, this analysis helps to understand expectations about home values.

• First, the data is collected from the sample population, in this case, 800 homeowners.
• Then, responses are categorized into distinct groups such as 'decrease', 'stay the same', or varying degrees of 'increase'.
• The purpose of analyzing this data is not just to record answers, but to predict the general sentiment or expectation among the population surveyed.

Data must be carefully interpreted to ensure valid conclusions and effective decision-making.

Fraction Simplification

Fraction simplification is a method used to express a fraction in its simplest form. This makes it easier to understand and work with. For instance, if we have a fraction like \( \frac{200}{800} \), it can initially seem bulky or complex. To simplify:

• Determine the greatest common divisor (GCD) of the numerator and the denominator. Here, the GCD of 200 and 800 is 200.
• Divide both the numerator and the denominator by their GCD: \( \frac{200}{800} = \frac{200 \div 200}{800 \div 200} = \frac{1}{4} \).

Simplifying fractions is an essential skill for quickly understanding proportions and making calculations easier. It allows the fraction to be interpreted more intuitively, such as \( \frac{1}{4} \) being equivalent to one-quarter.

Percentages

Percentages are a way to express a number as a fraction of 100. They are commonly used to compare relative sizes of quantities or to describe proportions. In probability problems, converting fractions to percentages is useful for understanding how often a specific outcome might occur.

• To convert a fraction, such as \( \frac{1}{4} \), into a percentage, multiply by 100. This gives: \( \frac{1}{4} \times 100 = 25\% \).
• For another example, the fraction \( \frac{23}{50} \) when multiplied by 100 equals \( 46\% \).

By expressing fractions as percentages, we can easily visualize probability outcomes and compare different scenarios. It's an important step in making sense of numerical data.

Statistics

Statistics involve the collection, analysis, interpretation, and presentation of data. They are used to derive insights and make informed decisions based on data. In the survey exercise, statistics help us quantify the homeowners' expectations about property value changes.

• Firstly, data is collected from a sample of 800 people.
• The data is then analyzed to find out how many expect values to decrease, stay the same, or increase.
• Using statistical analysis, probabilities are assigned to these expectations, offering a numerical representation of each outcome's likelihood.

Statistics provide a foundation for understanding trends and predicting future events based on past data. This technique is crucial for navigating decisions where uncertainty exists.