/*! 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 17 Consider the multivariable linea... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 17

Consider the multivariable linear function $$y=78.1+0.83 x_{1}-0.09 x_{2}+1.19 x_{3}$$ Evaluate this function for the given explanatory values. $$x_{1}=63, x_{2}=21, x_{3}=17$$

Short Answer

Expert verified
The value of \(y\) is 148.73.

Step by step solution

01

Substitute Values into the Function

First, we need to substitute the given values of \(x_1\), \(x_2\), and \(x_3\) into the function. The values are \(x_1 = 63\), \(x_2 = 21\), and \(x_3 = 17\). The original function is \(y = 78.1 + 0.83x_1 - 0.09x_2 + 1.19x_3\). Substituting the values, we get:\[y = 78.1 + 0.83(63) - 0.09(21) + 1.19(17)\]
02

Calculate Each Term

Now, calculate each term separately:- The first term is \(78.1\).- The second term is \(0.83 \times 63 = 52.29\).- The third term is \(-0.09 \times 21 = -1.89\).- The fourth term is \(1.19 \times 17 = 20.23\).
03

Sum All the Terms

Add all the calculated terms together to obtain the value of \(y\):\[y = 78.1 + 52.29 - 1.89 + 20.23\]
04

Final Calculation

Compute the sum from the previous step to find the value of \(y\).\[y = 148.73\]Thus, the value of \(y\) when \(x_1 = 63\), \(x_2 = 21\), and \(x_3 = 17\) is 148.73.

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.

Explanatory Variables
In the realm of multivariable linear functions, explanatory variables are those variables that help to explain variations or patterns we observe in the dependent variable, often denoted as \(y\). Simply put, they are inputs that influence the output, and in this case, they are represented by \(x_1\), \(x_2\), and \(x_3\) in the function.
Let's break it down:
  • \(x_1\), \(x_2\), and \(x_3\) are distinct characteristics or metrics that have some impact on the result \(y\).
  • Each explanatory variable is multiplied by a coefficient, representing its magnitude of influence.
  • These coefficients can be positive or negative, indicating whether an increase in the explanatory variable increases or decreases \(y\).
Therefore, understanding these variables is crucial to understand how changes in inputs affect outcomes, as shown in our function: \(y = 78.1 + 0.83x_1 - 0.09x_2 + 1.19x_3\).
Each term in the function reveals the potential change in \(y\) given a unit change in the corresponding explanatory variable.
Linear Equations in Statistics
Linear equations in statistics are foundational in explaining relationships between variables. They serve as a predictive tool, allowing us to anticipate the behavior of one variable based on known changes in another. This particularly applies to our function, where
  • The function format \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3\) is typical in linear models.
  • The term \(\beta_0\) is the intercept, signifying where the line crosses the y-axis. It's the value of \(y\) when all \(x\) terms are zero.
  • Each \(\beta\) associated with \(x\) is a slope, showing how much \(y\) changes for a one-unit increase in the explanatory variable.
These equations model real-world data, making it easier to understand trends and make predictions.
By fitting a linear equation to data, we can identify relationships and potential causal effects, providing insights grounded in statistical reality. This is key in fields widespread, from economics to engineering.
Calculating Function Values
Calculating function values is a practical process of plugging values into the equation and solving to find the outcome鈥攖he dependent variable \(y\). To calculate it accurately, follow these steps:
  • Start with the function equation, given by \(y = 78.1 + 0.83x_1 - 0.09x_2 + 1.19x_3\).
  • Substitute the provided values: \(x_1 = 63\), \(x_2 = 21\), \(x_3 = 17\).
  • Perform multiplication for each explanatory variable: calculate \(0.83 \times 63\), \(-0.09 \times 21\), \(1.19 \times 17\).
  • Add the results, starting with a constant, \(78.1\), modified by the outcomes of these multiplications.
This step-by-step calculation leads us to find \(y = 148.73\), providing a specific value for \(y\) based on given input values.
Understanding this process helps demystify how changes in inputs trigger changes in output鈥攁 fundamental aspect of analyzing linear functions.

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

Consider the following scenario: A random sample of 12 patients had their age, \(x_{1},\) and weight (in pounds), \(x_{2},\) recorded. Then their blood pressure was taken. It was found that their systolic blood pressure reading (in \(\mathrm{mm} \mathrm{Hg}\) ), \(y,\) can be modeled by the function $$y=30.99+0.86 x_{1}+0.33 x_{2}$$ The actual systolic blood pressure reading for a 64 -year-old person weighing 196 pounds was \(154 \mathrm{mm}\) Hg. Calculate the absolute error.

Determine which of the functions represent multivariable linear functions. $$y=1-6 x_{1}+11 x_{2}$$

Consider the following information: Political and social unrest, coupled with fluctuating fuel prices and shorter vacation times, have resulted in a leveling off of travel to foreign regions by Americans. The number of Americans traveling to Middle Eastern countries from 2001 to 2016 can be modeled by the logarithmic function $$f(x)=42.1+546.5 \ln x \quad 1 \leq x \leq 16$$ where \(x\) represents the number of years since 2000 and \(f(x)\) represents the number o Americans who traveled to Middle Eastern countries annually, measured in thousands Evaluate \(f(1)\) and interpret the result.

Consider the following information: The death rate in a community is attributed to several factors, such as doctor and hospital availability, income, and access to medical care. This can be modeled by the function $$y=12.57-0.01 x_{1}-0.36 x_{2}-0.01 x_{3}$$ where \(x_{1}\) represents the number of doctors available per 100,000 residents, \(x_{2}\) represents the annual per capita income in thousands of dollars, \(x_{3}\) is the population density per square mile, and \(y\) represents the death rate per 100,000 individuals. Determine the death rate when the doctor availability is 68 doctors per 100,000 residents, the annual per capita income is 8,700 dollar and the population density is 144 per square mile. Round the answer to one decimal place.

A rational exponent function is given. Evaluate the function at the indicated value, then graph the function for the specified independent variable values. Round the function values to two decimal places as necessary. \(f(x)=x^{\frac{10}{9}} ;\) Evaluate \(f(0), f(10), f(20) .\) Graph \(f(x)\) for \(0 \leq x \leq 30\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.