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Chapter 1: Problem 36

For each data set, write a model for the data as given and a model for the inverted data. The table gives the percentage of companies that are still in operation \(t\) years after they first start. Business Survival (years after beginning operation) $$ \begin{array}{|c|c|} \hline \text { Years } & \begin{array}{c} \text { Companies } \\ \text { (percentage) } \end{array} \\ \hline 5 & 50 \\ \hline 6 & 47 \\ \hline 7 & 44 \\ \hline 8 & 41 \\ \hline 9 & 38 \\ \hline 10 & 35 \\ \hline \end{array} $$

Short Answer

Expert verified
Model: \( P = -3t + 65 \); Inverted Model: \( t = \frac{65 - P}{3} \).

Step by step solution

01

Understand the provided data

The data shows how the percentage of companies that survive decreases with each passing year. This can be modeled as a linear function since the percentage drops steadily.
02

Define variables for the model

Let \( t \) represent the years after beginning operation, and let \( P \) represent the percentage of companies still in operation. We need to find a relationship between \( P \) and \( t \).
03

Determine the linear model

We observe that for every increase of 1 year in \( t \), \( P \) decreases by 3%. Using the point-slope form of a linear equation, we can set up our linear model based on the points (5, 50) and (10, 35).
04

Use point-slope formula

The slope \( m \) is \(-3 \) (since \( m = \frac{35 - 50}{10 - 5} = -3 \)). Using point-slope form \( P = m(t - t_1) + P_1 \), where \( t_1 = 5 \) and \( P_1 = 50 \), our model becomes \( P = -3(t - 5) + 50 \). Simplifying gives \( P = -3t + 65 \).
05

Define inverted data variables

For the inverted data, swap \( P \) and \( t \). Here, \( t \) becomes a function of \( P \), meaning we need to express \( t \) in terms of \( P \).
06

Invert the linear model

Using the previous model \( P = -3t + 65 \), solve for \( t \). Rearranging gives \( t = \frac{65 - P}{3} \). This inverted model describes years of operation as a function of the percentage still in operation.

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Key Concepts

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

Data modeling
Data modeling is essential in data analysis, as it helps in creating a simplified representation of real-world data. In the context of the provided exercise, data modeling involves representing the survival of companies over time through a mathematical model.

Here, we are concerned with business survival statistics, showing how the percentage of operational companies changes each year after they start.
  • A data model using a linear function helps you understand trends in the data.
  • This model allows for predictions about company survival rates beyond the observed period.
By summarizing complex data into a simple equation, one can easily analyze trends and make informed decisions. Through this approach, businesses can strategize better to improve survival rates.
Linear function
A linear function is used when there is a constant rate of change between two variables. In our exercise, the relationship between years of operation and the percentage of companies still in business is linear.

The key characteristics of a linear function include:
  • It has a straight-line graph.
  • The equation format is generally in the form of: \[ y = mx + c \] where \( m \) is the slope and \( c \) is the y-intercept.
  • It shows a consistent degree of change.
In the exercise, years (\( t \)) and company survival percentage (\( P \)) are linearly related, with the linear equation derived being \[ P = -3t + 65 \]. This implies that with each passing year, the percentage of surviving companies decreases by 3%.
Point-slope form
The point-slope form is a technique used to write the equation of a line when you have one point and the slope. It is particularly useful when fitting a linear model to observed data points.

The general form is \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is a known point on the line, and \( m \) is the slope.
  • In this case, our known points were (5, 50) and (10, 35).
  • The slope \( m \) calculated as -3, indicating a drop of 3% for each additional year.
  • By substituting into the formula, we derived the equation \[ P = -3(t - 5) + 50 \].
Finally, after simplification, the equation becomes \[ P = -3t + 65 \], providing a clear model for prediction and analysis.
Inverted data analysis
Inverted data analysis provides a different perspective by reversing the usual roles of the dependent and independent variables. It involves expressing one variable as a function of the other.

For the exercise at hand, inverting the data means expressing years in terms of the percentage of companies still in operation (i.e., solving for \( t \) given \( P \)).
  • Starting with the linear model \[ P = -3t + 65 \], solve for \( t \).
  • Rearrange to get \[ t = \frac{65 - P}{3} \].
  • This inverted model helps answer questions like: "For a given survival percentage, how many years of operation does this represent?"
Inverted analysis is useful when you need to determine the input condition (years) needed to achieve a specific outcome (survival percentage).

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Most popular questions from this chapter

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