/*! 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 33 Exercises 33 and 34 refer to the... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 33

Exercises 33 and 34 refer to the following setting. Thirty randomly selected seniors at Council High School were asked to report the age (in years) and mileage of their main vehicles. Here is a scatterplot of the data: We used Minitab to perform a least-squares regression analysis for these data. Part of the computer output from this regression is shown below. Predictor \(\quad\) coef \(\quad\) stdev \(\quad\) t-ratio \(\quad \mathrm{P}\) Constant \(-13832 \qquad 8773 \qquad-1.58 \qquad 0.126\) Age \(\quad 14954 \qquad 1546 \qquad 9.67 \quad 0.000\) \(s=22723 \qquad R-s q=77.08 \qquad R-s q(a d j)=76.18\) Drive my car (3.2) (a) What is the equation of the least-squares regression line? Be sure to define any symbols you use. (b) Interpret the slope of the least-squares line in the context of this problem. (c) One student reported that her 10-year-old car had 110,000 miles on it. Find the residual for this data value. Show your work.

Short Answer

Expert verified
(a) \( \hat{y} = -13832 + 14954x \); (b) Slope means mileage increases by 14954 per year; (c) Residual: -25708 miles.

Step by step solution

01

Identify the Regression Equation

From the given regression output, we can identify the coefficients provided for the constant and the age predictor. This allows us to form the equation of the least-squares regression line. The general form of the regression line equation is given by \( y = b_0 + b_1x \) where \( y \) is the predicted mileage, \( b_0 \) is the y-intercept, and \( b_1 \) is the slope. Here, the constant \( (b_0) \) is -13832 and the slope \( (b_1) \) for age is 14954. So, the regression equation is: \[ \hat{y} = -13832 + 14954x \] Where \( x \) represents the age of the vehicle in years, and \( \hat{y} \) represents the predicted mileage.
02

Interpret the Slope

The slope of the regression line, 14954, represents the change in the predicted mileage for each additional year of the vehicle's age. In the context of this problem, the slope means that for each additional year a vehicle ages, the predicted mileage increases by 14954 miles. This suggests that older cars tend to have higher mileage.
03

Calculate the Predicted Mileage for a 10-Year-Old Car

To find the residual, we first need to compute the predicted mileage for the 10-year-old car using the regression equation. Substitute \( x = 10 \) into the regression equation:\[ \hat{y} = -13832 + 14954(10) \] Calculating this, we find: \[ \hat{y} = -13832 + 149540 = 135708 \] Thus, the predicted mileage for a 10-year-old car is 135708 miles.
04

Calculate the Residual

The residual is the difference between the actual mileage and the predicted mileage. Given that the actual mileage is 110,000 miles, we find the residual by subtracting the predicted mileage from the actual mileage:\[ ext{Residual} = ext{Actual Mileage} - ext{Predicted Mileage} = 110000 - 135708 = -25708 \] The residual for this data point is -25708 miles, which indicates that the car has 25708 fewer miles than predicted by the model.

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Key Concepts

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

scatterplot analysis
A scatterplot is a visual representation that displays the relationship between two quantitative variables. In the context of Council High School's dataset, each point on the scatterplot represents a senior's vehicle, showing the age of the vehicle (in years) on the horizontal axis and its mileage on the vertical axis.
Scatterplots are a fantastic way to start any regression analysis because they help us quickly grasp the general direction and strength of the relationship between variables. You'll typically look for a pattern or trend in the data points, which could indicate a correlation.
  • If the points trend upwards, there might be a positive correlation, indicating that as one variable increases, the other does as well.
  • If they trend downwards, it might suggest a negative correlation, where one variable decreases as the other increases.
  • If the points are scattered with no discernible pattern, there's likely no correlation.
This particular scatterplot analysis allows us to see how mileage tends to increase as the age of the vehicles increases, setting the stage for least-squares regression analysis.
residual calculation
Residuals are one of the crucial metrics in regression analysis. They help us measure the deviation of the observed values from the values predicted by the regression line. A residual is calculated as follows: \[ \text{Residual} = \text{Actual Value} - \text{Predicted Value} \]To calculate the residual for a specific data point, say a car that's 10 years old with a mileage of 110,000, you follow these steps:
  • First, use the regression equation to find the predicted mileage for a 10-year-old car. In our case, this equation is \[ \hat{y} = -13832 + 14954x \]
    By substituting 10 for the vehicle's age, we get a predicted mileage of 135,708.
  • Then, subtract this predicted mileage from the actual mileage, which is 110,000 in this scenario.
The residual, therefore, is \[ 110,000 - 135,708 = -25,708 \], indicating that the car has driven 25,708 fewer miles than the model predicted. Negative residuals mean the prediction overestimated the mileage.
slope interpretation
The slope in a regression equation often offers significant insights into the relationship between variables. When we interpret the slope, especially in a real-world context like our dataset, it tells us how the dependent variable (mileage) is expected to change as the independent variable (age) increases by one unit.
In our regression model, the slope, represented by the coefficient 14,954, means that for every additional year the vehicle ages, the predicted mileage increases by 14,954 miles. This positive value suggests a direct relationship between vehicle age and its accumulated mileage. In simple terms, older cars generally have higher mileage.
Here are a few things to remember about slope interpretation:
  • A positive slope indicates a positive relationship, meaning both variables tend to move in the same direction.
  • A negative slope would indicate an inverse relationship.
  • The magnitude of the slope determines how steeply one variable affects the other.
Understanding slope is critical for students analyzing a scatterplot because it encapsulates the essence of the relationship depicted and offers an easy-to-grasp summary of the data you鈥檙e dealing with.

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Most popular questions from this chapter

In Exercises 39 to 42, determine whether or not the conditions for using two- sample t procedures are met. Literacy rates Do males have higher average literacy rates than females in Islamic countries? The table below shows the percent of men and women at least 15 years old who were literate in 2008 in the major Islamic nations. (We omitted countries with populations of less than 3 million.) Data for a few nations, such as Afghanistan and Iraq, were not available.\(^{30}\) $$\begin{array}{lll}{\text { Country }} & {\text { Female percent }} & {\text { Male percent }} \\ {\text { Algeria }} & {66} & {94} \\ {\text { Bangladesh }} & {48} & {71} \\ {\text { Egypt }} & {58} & {88} \\ {\text { lran }} & {77}& {97} \\ {\text { Jordan }} & {87} & {99} \\ {\text { Kazakhstan }} & {100} & {100}\\\\{\text { Lebanon }} & {86} & {98} \\ {\text { Libya }} & {78} & {100}\\\\{\text { Libya }} & {78} & {100} \\ {\text { Malaysia }} & {90} & {98}\\\ {\text { Morocoo }} & {43}& {84} \\ {\text { Saudi Arabia }} & {79} & {98}\\\\{\text { Syria }} & {77} & {95} \\ {\text { Taijkistan }} & {100} & {100} \\ {\text { Tunisia }} & {69} & {97}\\\\{\text { Turkey }} & {81}& {99} \\ {\text { Uzbekistan }} & {96} & {99} \\ {\text { Yemen }} & {41} & {93}\end{array}$$

Is red wine better than white wine? Observational studies suggest that moderate use of alcohol by adults reduces heart attacks and that red wine may have special benefits. One reason may be that red wine contains polyphenols, substances that do good things to cholesterol in the blood and so may reduce the risk of heart attacks. In an experiment, healthy men were assigned at random to drink half a bottle of either red or white wine each day for two weeks. The level of polyphenols in their blood was measured before and after the two-week period. Here are the percent changes in level for the subjects in both groups:\(^{31}\) Red wine: 3.58 .1\(\quad 7.44 .0 \quad 0.74 .98 .4 \quad 7.05 .5\) White wine: \(3.1 \quad 0.5-3.8\) 4.1 \(-0.62 .7 \quad 1.9-5.9 \quad 0.1\) (a) A Fathom dotplot of the data is shown below. Use the graph to answer these questions: \(\bullet\) Are the centers of the two groups similar or different? Explain. \(\bullet\) Are the spreads of the two groups similar or different? Explain. (b) Construct and interpret a 90% confidence interval for the difference in mean percent change in polyphenol levels for the red wine and white wine treatments. (c) Does the interval in part (b) suggest that red wine is more effective than white wine? Explain.

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