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Chapter 13: Problem 15

Plot the following straight lines. Give the values of the \(y\) -intercept and slope for each of these lines and interpret them. Indicate whether each of the lines gives a positive or a negative relationship between \(x\) and \(y\) a. \(y=100+5 x \quad\) b. \(y=400-4 x\)

Short Answer

Expert verified
For the equation \(y=100+5x\), the slope is 5 (indicating a positive relationship between \(x\) and \(y\)) and the \(y\)-intercept is 100. For the equation \(y=400-4x\), the slope is -4 (indicating a negative relationship between \(x\) and \(y\)) and the \(y\)-intercept is 400.

Step by step solution

01

Plot the line representing equation a

In the equation \(y=100+5x\), the coefficient of \(x\) is the slope of the line which is 5, and the constant term is the \(y\)-intercept. Therefore, the slope of the line is 5 and the \(y\)-intercept is 100. Given that the slope is positive, the line moves upward from left to right, indicating a positive correlation between \(x\) and \(y\). Plot this line on a graph.
02

Interpretation of equation a

Given that the slope is 5, it means that for an increase of 1 unit in \(x\), \(y\) increases by 5 units. The \(y\)-intercept being 100 tells us that when \(x\) equals 0, \(y\) equals 100.
03

Plot the line representing equation b

In the equation \(y=400-4x\), the coefficient of \(x\) is the slope of the line which is -4, and the constant term is the \(y\)-intercept. Therefore, the slope of the line is -4 and the \(y\)-intercept is 400. Given that the slope is negative, the line moves downward from left to right, indicating a negative correlation between \(x\) and \(y\). Plot this line on a graph.
04

Interpretation of equation b

Given that the slope is -4, it means that for an increase of 1 unit in \(x\), \(y\) decreases by 4 units. The \(y\)-intercept being 400 tells us that when \(x\) equals 0, \(y\) equals 400.

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Key Concepts

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

Slope Interpretation
The **slope** of a linear equation represents the rate of change between the two variables, `x` and `y`.It can be thought of as the 'steepness' of the line when graphed.
If the slope is positive, like in the equation \(y=100+5x\), it means the line rises as you move from left to right. In practical terms, for every unit increase in `x`, `y` increases by 5 units.
Conversely, a negative slope indicates that the line falls as you move from left to right. For instance, the equation \(y=400-4x\) has a slope of -4. This means that for every 1 unit increase in `x`, `y` decreases by 4 units.
  • A positive slope implies a direct relationship between `x` and `y`: as `x` increases, so does `y`.
  • A negative slope implies an inverse relationship: as `x` increases, `y` decreases.
Y-Intercept
The **y-intercept** of a line is the point at which it crosses the `y`-axis.This point is crucial because it tells you the value of `y` when `x` is zero.
For the equation \(y=100+5x\), the y-intercept is 100. This means that when no x-values are considered (i.e., `x` is zero), `y` will begin at 100.
Similarly, the equation \(y=400-4x\) has a y-intercept of 400, meaning `y` will be 400 when `x` is zero.
  • The y-intercept provides a starting point for the line on a graph.
  • It's where the change modeled by `x` begins.
Positive and Negative Correlation
**Correlation** refers to the relationship between two variables.In linear equations, this is demonstrated by the slope of the line.
- A **positive correlation** indicates that as one variable increases, the other variable follows the same trend. This is evidenced by an upward slope, as seen in the equation \(y=100+5x\). Here, both `x` and `y` increase together.
- A **negative correlation** shows an opposite trend. As one variable increases, the other decreases. The equation \(y=400-4x\) exemplifies this with its downward slope. As `x` increases, `y` decreases.
  • It is crucial in data analysis to determine the type of correlation to understand the relationship between variables.
  • Graphically, you can tell by simply looking at the direction of the line.
Graphing Linear Equations
**Graphing** linear equations involves plotting `x` and `y` values on a cartesian coordinate system to visualize the relationship.For linear equations, this always results in a straight line.
- To graph the equation \(y=100+5x\), start by plotting the y-intercept at (0, 100) on the `y`-axis.Then, use the slope to determine other points鈥攆or every 1 unit right (along the `x`-axis), move 5 units up.Draw a line through these points to complete the graph.
- For \(y=400-4x\), plot the y-intercept at (0, 400). Here, the slope indicates moving 4 units down for every 1 unit right.Similarly, connect these points to form the line.
  • Graphing helps visualize and interpret the real-world meaning of equations.
  • It shows how quickly changes occur between variables.

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Most popular questions from this chapter

A population data set produced the following information. $$ \begin{aligned} &N=250, \quad \Sigma x=9880, \quad \Sigma y=1456, \quad \Sigma x y=85,080 \\ &\Sigma x^{2}=485,870, \text { and } \Sigma y^{2}=135,675 \end{aligned} $$ Find the linear correlation coefficient \(\rho\).

Plot the following straight lines. Give the values of the \(y\) -intercept and slope for each of these lines and interpret them. Indicate whether each of the lines gives a positive or a negative relationship between \(x\) and \(y\). a. \(y=-60+8 x \quad\) b. \(y=300-6 x\)

The management of a supermarket wants to find if there is a relationship between the number of times a specific product is promoted on the intercom system in the store and the number of units of that product sold. To experiment, the management selected a product and promoted it on the intercom system for 7 days. The following table gives the number of times this product was promoted each day and the number of units sold. $$ \begin{array}{cc} \hline \begin{array}{c} \text { Number of Promotions } \\ \text { per Day } \end{array} & \begin{array}{c} \text { Number of Units Sold } \\ \text { per Day (hundreds) } \end{array} \\ \hline 15 & 11 \\ 22 & 22 \\ 42 & 30 \\ 30 & 26 \\ 18 & 17 \\ 12 & 15 \\ 38 & 23 \\ \hline \end{array} $$ a. With the number of promotions as an independent variable and the number of units sold as a dependent variable, what do you expect the sign of \(B\) in the regression line \(y=A+B x+\epsilon\) will be? b. Find the least squares regression line \(\hat{y}=a+b x\). Is the sign of \(b\) the same as you hypothesized for \(B\) in part a? c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Compute \(r\) and \(r^{2}\) and explain what they mean. e. Predict the number of units of this product sold on a day with 35 promotions. f. Compute the standard deviation of errors. g. Construct a \(98 \%\) confidence interval for \(B\). h. Testing at the \(1 \%\) significance level, can you conclude that \(B\) is positive? i. Using \(\alpha=.02\), can you conclude that the correlation coefficient is different from zero?

The following table gives information on the incomes (in thousands of dollars) and charitable contributions (in hundreds of dollars) for the last year for a random sample of 10 households. $$ \begin{array}{rc} \hline \text { Income } & \text { Charitable Contributions } \\ \hline 76 & 15 \\ 57 & 4 \\ 140 & 42 \\ 97 & 33 \\ 75 & 5 \\ 107 & 32 \\ 65 & 10 \\ 77 & 18 \\ 102 & 28 \\ 53 & 4 \\ \hline \end{array} $$ a. With income as an independent variable and charitable contributions as a dependent variable, compute \(\mathrm{SS}_{x \mathrm{x}}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x v}\) b. Find the regression of charitable contributions on income. c. Briefly explain the meaning of the values of \(a\) and \(b\). d. Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Test at the \(1 \%\) significance level whether \(B\) is positive. h. Using the \(1 \%\) significance level, can you conclude that the linear correlation coefficient is different from zero?

An economist is studying the relationship between the incomes of fathers and their sons or daughters. Let \(x\) be the annual income of a 30 -year-old person and let \(y\) be the annual income of that person's father at age 30 years, adjusted for inflation. A random sample of 300 thirty-year-old and their fathers yields a linear correlation coefficient of \(.60\) between \(x\) and \(y .\) A friend of yours, who has read about this research, asks you several questions, such as: Does the positive value of the correlation coefficient suggest that the 30 -year-old tend to earn more than their fathers? Does the correlation coefficient reveal anything at all about the difference between the incomes of 30 -year-old and their fathers? If not, what other information would we need from this study? What does the correlation coefficient tell us about the relationship between the two variables in this example? Write a short note to your friend answering these questions.
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