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Chapter 3: Problem 6

Driving: You are driving on a highway. The following table gives your speed \(S\), in miles per hour, as a function of the time \(t\), in seconds, since you started making your observations. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Time } t & 0 & 15 & 30 & 45 & 60 \\ \hline \text { Speed S } & 54 & 59 & 63 & 66 & 68 \\ \hline \end{array} $$ a. Find the equation of the regression line that expresses \(S\) as a linear function of \(t\). b. Explain in practical terms the meaning of the slope of the regression line. c. On the basis of the regression line model, when do you predict that your speed will reach 70 miles per hour? (Round your answer to the nearest second.) d. Plot the data points and the regression line. e. Use your plot in part \(d\) to answer the following: Is your prediction in part c likely to give a time earlier or later than the actual time when your speed reaches 70 miles per hour?

Short Answer

Expert verified
Equation: \( S = 0.24t + 54.8 \). Slope: Speed increases by 0.24 mph per second. Prediction: 63 seconds.

Step by step solution

01

Calculate Means

Calculate the mean of the given time values \( t \) and speed values \( S \). \[ \bar{t} = \frac{0 + 15 + 30 + 45 + 60}{5} = 30 \] \[ \bar{S} = \frac{54 + 59 + 63 + 66 + 68}{5} = 62 \] These means will help in determining the slope and y-intercept of the regression line.
02

Calculate the Slope of the Regression Line

Use the formula for the slope \( m \) of the regression line: \[ m = \frac{\sum{((t_i - \bar{t})(S_i - \bar{S}))}}{\sum{(t_i - \bar{t})^2}} \] Calculate the terms: \[ \sum{((t_i - \bar{t})(S_i - \bar{S}))} = (0-30)(54-62) + (15-30)(59-62) + (30-30)(63-62) + (45-30)(66-62) + (60-30)(68-62) = 540 \]\[ \sum{(t_i - \bar{t})^2} = (0-30)^2 + (15-30)^2 + (30-30)^2 + (45-30)^2 + (60-30)^2 = 2250 \]Thus, \[ m = \frac{540}{2250} = 0.24 \].
03

Calculate the Y-Intercept of the Regression Line

Use the formula for the y-intercept \( b \) of the regression line: \[ b = \bar{S} - m \times \bar{t} \] \[ b = 62 - 0.24 \times 30 = 54.8 \].
04

Write the Regression Equation

The equation of the regression line is: \[ S = 0.24t + 54.8 \], where \( S \) is the speed in mph and \( t \) is the time in seconds.
05

Interpret the Slope

The slope \( 0.24 \) indicates the rate at which the speed increases per second. Practically, for every additional second, the speed increases by 0.24 mph.
06

Predict Time to Reach 70 mph

Use the regression equation to solve for \( t \) when \( S = 70 \):\[ 70 = 0.24t + 54.8 \]Solving for \( t \):\[ t = \frac{70 - 54.8}{0.24} \approx 63.33 \]Round this to the nearest second: \( t \approx 63 \) seconds.
07

Plot Data and Regression Line

Plot the given data points \((t, S)\) and the regression line \( S = 0.24t + 54.8 \) on the graph. The data points will show slight deviations but should roughly align with the regression line.
08

Evaluating Prediction Accuracy

Analyzing the plot, because the points are not exactly on the straight line, the regression line provides an average fit. The prediction might slightly differ from the actual time your speed reaches 70 mph.

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Key Concepts

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

Slope Interpretation
The slope in the context of a linear regression line is a crucial element. It quantifies the rate of change in the dependent variable as the independent variable changes. In this exercise, the slope is calculated to be 0.24. What does this mean? It means that for every second that passes, your speed increases by 0.24 miles per hour.

This small increment might seem insignificant, but over several seconds, it can lead to substantial changes in speed. This interpretation is essential for understanding how the dependent variable, speed in this scenario, responds to the time variable. It helps in predicting the behavior of speed over time.
Regression Line Equation
The regression line equation represents the relationship between variables in a linear form. It is expressed as: \[ S = 0.24t + 54.8 \] Here, \( S \) is the speed while \( t \) is the time in seconds. The formula reveals how speed can be predicted for any given time.

The key components of this equation are the slope (0.24) and the y-intercept (54.8). The y-intercept indicates the starting speed at the time zero. Starting at 54.8 mph suggests that, at the moment observations began, the initial speed was already at this point. Together, these components form the backbone of the linear equation that models the data.
Prediction Accuracy
Prediction accuracy indicates how well the regression line represents the actual data points. In this instance, the regression line predicts when the speed will reach 70 mph.

By solving the regression equation, we estimate that 70 mph is reached when \( t \) is approximately 63 seconds. However, because real-world data often contains variability, no regression line will fit all points perfectly.

This predicted value is a best estimate given the data and may slightly differ from actual results. Variability can be attributed to factors not accounted for in the simple linear model鈥攁n important consideration in evaluating the reliability of predictions.
Data Plotting
Data plotting involves graphing the observed data points and the regression line to visually assess how well the line fits the data. In our scenario, plotting time against speed reveals how the regression line, \( S = 0.24t + 54.8 \), approximates the dataset.

Each point represents a coordinate of time and speed. Observing how closely these points align with the line helps understand the model's accuracy.
  • If points lie along or near the line, the model performs well in describing the data.
  • Points that deviate significantly suggest areas where the model may not capture all trends accurately.


Using plots is invaluable; it shows any discrepancies and reinforces understanding by providing a visual context to the mathematical analysis.

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Most popular questions from this chapter

Total cost: The total cost \(C\) for a manufacturer during a given time period is a function of the number \(N\) of items produced during that period. To determine a formula for the total cost, we need to know two things. The first is the manufacturer's fixed costs. This amount covers expenses such as plant maintenance and insurance, and it is the same no matter how many items are produced. The second thing we need to know is the cost for each unit produced, which is called the variable cost. Suppose that a manufacturer of widgets has fixed costs of \(\$ 1500\) per month and that the variable cost is \(\$ 20\) per widget (so it costs \(\$ 20\) to produce 1 widget). a. Explain why the function giving the total monthly cost \(C\), in dollars, of this widget manufacturer in terms of the number \(N\) of widgets produced in a month is linear. Identify the slope and initial value of this function, and write down a formula. b. Another widget manufacturer has a variable cost of \(\$ 12\) per widget, and the total cost is \(\$ 3100\) when 150 widgets are produced in a month. What are the fixed costs for this manufacturer? c. Yet another widget manufacturer has determined the following: The total cost is \(\$ 2700\) when 100 widgets are produced in a month, and the total cost is \(\$ 3500\) when 150 widgets are produced in a month. What are the fixed costs and variable cost for this manufacturer?

Growth in height: Between the ages of 7 and 11 years, a certain boy grows 2 inches taller each year. At age 9 he is 48 inches tall. a. Explain why, during this period, the function giving the height of the boy in terms of his age is linear. Identify the slope of this function. b. Use a formula to express the height of the boy as a linear function of his age during this period. Be sure to identify what the letters that you use mean. c. What is the initial value of the function you found in part \(b\) ? d. Studying a graph of the boy's height as a function of his age from birth to age 7 reveals that the graph is increasing and concave down. Does this indicate that his actual height (or length) at birth was larger or smaller than your answer to part c? Be sure to explain your reasoning.

Where lines with different slopes meet: On the same coordinate axes, draw one line with vertical intercept 2 and slope 3 and another with vertical intercept 4 and slope 1. Do these lines cross? If so, do they cross to the right or left of the vertical axis? In general, if one line has its vertical intercept below the vertical intercept of another, what conditions on the slope will ensure that the lines cross to the right of the vertical axis?

Male and female high school graduates: The table below shows the percentage of male and female high school graduates who enrolled in college within 12 months of graduation. \({ }^{31}\) $$ \begin{array}{|l|c|c|c|c|} \hline \text { Year } & 1960 & 1965 & 1970 & 1975 \\ \hline \text { Males } & 54 \% & 57.3 \% & 55.2 \% & 52.6 \% \\ \hline \text { Females } & 37.9 \% & 45.3 \% & 48.5 \% & 49 \% \\ \hline \end{array} $$ a. Find the equation of the regression line for percentage of male high school graduates entering college as a function of time. b. Find the equation of the regression line for percentage of female high school graduates entering college as a function of time. c. Assume that the regression lines you found in part a and part b represent trends in the data. If the trends persisted, when would you expect first to have seen the same percentage of female and male graduates entering college? (You may be interested to know that this actually occurred for the first time in 1980 . The percentages fluctuated but remained very close during the \(1980 \mathrm{~s}\). In the 1990 s significantly more female graduates entered college than did males. In 1992 , for example, the rate for males was \(59.6 \%\) compared with \(63.8 \%\) for females.)

Energy cost of running: Physiologists have studied the steady-state oxygen consumption (measured per unit of mass) in a running animal as a function of its velocity (i.e., its speed). They have determined that the relationship is approximately linear, at least over an appropriate range of velocities. The table on the following page gives the velocity \(v\), in kilometers per hour, and the oxygen consumption \(E\), in milliliters of oxygen per gram per hour, for the rhea, a large, flightless South American bird. \({ }^{27}\) (For comparison, 10 kilometers per hour is about \(6.2\) miles per hour.) $$ \begin{array}{|c|c|} \hline \text { Velocity } v & \text { Oxygen consumption } E \\ \hline 2 & 1.0 \\ \hline 5 & 2.1 \\ \hline 10 & 4.0 \\ \hline 12 & 4.3 \\ \hline \end{array} $$ a. Find the equation of the regression line for \(E\) in terms of \(v\). b. The slope of the linear function giving oxygen consumption in terms of velocity is called the cost of transport for the animal, since it measures the energy required to move a unit mass by 1 unit distance. What is the cost of transport for the rhea? c. Physiologists have determined the general approximate formula \(C=8.5 \mathrm{~W}^{-0.40}\) for the cost of transport \(C\) of an animal weighing \(W\) grams. If the rhea weighs 22,000 grams, is its cost of transport from part b higher or lower than what the general formula would predict? Is the rhea a more or a less efficient runner than a typical animal its size? d. What would your equation from part a lead you to estimate for the oxygen consumption of a rhea at rest? Would you expect that estimate to be higher or lower than the actual level of oxygen consumption of a rhea at rest?
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