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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. \(y=100+5 x\) b. \(y=400-4 x\)

Short Answer

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

Step by step solution

01

Identify the Slope and Y-intercept for Line (a)

From the equation \(y = 100 + 5x\), the slope, \(m\), is 5 and \(y\)-intercept, \(c\), is 100.
02

Interpret and Define Relationship for Line (a)

Since the slope is positive, it shows a positive relationship between \(x\) and \(y\). As \(x\) increases, \(y\) increases. The \(y\)-intercept being 100 tells us that the line intersects the \(y\)-axis at the point (0, 100).
03

Identify the Slope and Y-intercept for Line (b)

From the equation \(y = 400 - 4x\), the slope, \(m\), is -4 and \(y\)-intercept, \(c\), is 400.
04

Interpret and Define Relationship for Line (b)

The negative slope indicates a negative relationship between \(x\) and \(y\), meaning as \(x\) increases, \(y\) decreases. The \(y\)-intercept being 400 means that the line intersects the \(y\)-axis at the point (0, 400).

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Most popular questions from this chapter

A researcher took a sample of 10 years and found the following relationship between \(x\) and \(y\), where \(x\) is the number of major natural calamities (such as tornadoes, hurricanes, earthquakes, floods, etc.) that occurred during a year and \(y\) represents the average annual total profits (in millions of dollars) of a sample of insurance companies in the United States. $$ \hat{y}=342.6-2.10 x a. A randomly selected year had 24 major calamities. What are the expected average profits of U.S. insurance companies for that year? b. Suppose the number of major calamities was the same for each of 3 years. Do you expect the average profits for all U.S. insurance companies to be the same for each of these 3 years? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact? $$

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While browsing through the magazine rack at a bookstore, a statistician decides to examine the relationship between the price of a magazine and the percentage of the magazine space that contains advertisements. The data collected for eight magazines are given in the following table. Here price is the dependent variable. $$ \begin{array}{l|rrrr} \hline \text { Percentage containing ads } & 37 & 43 & 58 & 49 \\ \hline \text { Price (\$) } & 5.50 & 6.95 & 4.95 & 5.75 \\ \hline \text { Percentage containing ads } & 70 & 28 & 65 & 32 \\ \hline \text { Price (\$) } & 3.95 & 8.25 & 5.50 & 6.75 \\ \hline \end{array} $$ a. Find the standard deviation of errors. b. Compute the coefficient of determination. What percentage of the variation in price is explained by the least squares regression of price on the percentage of magazine space containing ads? What percentage of this variation is not explained?

Why is the random error term included in a regression model?

The following data give the experience (in years) and monthly salaries (in hundreds of dollars) of nine randomly selected secretaries. $$ \begin{array}{l|rrrrrrrrr} \hline \text { Experience } & 14 & 3 & 5 & 6 & 4 & 9 & 18 & 5 & 16 \\ \hline \begin{array}{l} \text { Monthly } \\ \text { salary } \end{array} & 62 & 29 & 37 & 43 & 35 & 60 & 67 & 32 & 60 \\ \hline \end{array} $$ Construct a \(90 \%\) confidence interval for the mean monthly salary of all secretaries with 10 years of experience. Construct a \(90 \%\) prediction interval for the monthly salary of a randomly selected secretary with 10 years of experience.
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