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Chapter 22: Problem 2

Find the equation of the least-squares line for the given data. Graph the line and data points on the same graph. $$\begin{array}{l|r|r|r|r|r|r|r} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline y & 10 & 17 & 28 & 37 & 49 & 56 & 72 \end{array}$$

Short Answer

Expert verified
The least-squares line equation is \( y = -24.57 + 15.75x \).

Step by step solution

01

Understanding the Data

We have a dataset with points (1, 10), (2, 17), (3, 28), (4, 37), (5, 49), (6, 56), and (7, 72). Our goal is to find the equation of the least-squares line (line of best fit) for this data.
02

Calculating the Mean Values

Calculate the mean of the x-values: \[ \bar{x} = \frac{1+2+3+4+5+6+7}{7} = 4 \]Calculate the mean of the y-values: \[ \bar{y} = \frac{10+17+28+37+49+56+72}{7} = 38.43 \]
03

Calculating the Slope (b)

The formula for the slope (b) of the least-squares line is: \[ b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]Calculate each term and substitute the values:\[ b = \frac{(1-4)(10-38.43) + (2-4)(17-38.43) + .... + (7-4)(72-38.43)}{(1-4)^2 + (2-4)^2 + ... + (7-4)^2} = \frac{441}{28} = 15.75 \]
04

Calculating the Intercept (a)

The formula for the intercept (a) is: \[ a = \bar{y} - b\bar{x} \]Substitute the known values:\[ a = 38.43 - (15.75)(4) = -24.57 \]
05

Forming the Equation

The equation of the least-squares line is given by: \[ y = a + bx \]Substitute a and b:\[ y = -24.57 + 15.75x \]
06

Graphing the Line and Data Points

Plot the data points on a graph and the least-squares line using the equation \( y = -24.57 + 15.75x \). The line should pass through and fit the overall trend of the data points.

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Key Concepts

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

Line of Best Fit
A line of best fit, also known as a least-squares line, is a straight line that best represents the data on a scatter plot. It minimizes the total distance between the points and the line, capturing the trend of the given data points.
  • When you plot data on a scatter plot, you might notice that the points seem to go upward, downward, or even form a more complex pattern. The line of best fit shows the general direction (or trend) that the data is moving.
  • When finding a line of best fit, our goal is to predict the behavior of the dependent variable based on the independent variable.
To create the line of best fit for the given dataset, both the slope and the y-intercept need to be calculated using statistical methods, specifically using least-squares regression. This involves a bit of number crunching where we perform calculations to minimize the differences between observed and predicted values.
Slope Calculation
The slope of the least-squares line represents how much the y-value increases or decreases for each unit increase in the x-value. Calculating the slope is a key step in determining the line of best fit.
  • The formula to calculate the slope (often denoted as \( b \)) is essential. It's given by: \[ b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
  • This formula is applied by taking each x-value, subtracting the mean of x, and multiplying it by the difference of each y-value from the mean of y, then summing up these products.
  • After this, we divide by the sum of squared differences of each x-value from the mean of x to complete the formula.
For our specific data, the slope is calculated to be 15.75, which means for every increase of 1 in the x-value, the y-value is expected to increase by 15.75.
Intercept Calculation
The intercept of the least-squares line is the point where the line crosses the y-axis. It gives us the starting value of the dependent variable when the independent variable is 0.
  • Here, we use the formula:\[a = \bar{y} - b\bar{x} \]
  • This involves taking the mean of the y-values and subtracting the product of the slope and the mean of the x-values.
  • By plugging in our previous results, \( \bar{y} = 38.43 \) and \( \bar{x} = 4 \), along with the slope \( b = 15.75 \), we find the intercept to be \( a = -24.57 \).
This intercept tells us that when x is equal to zero, the y-value would theoretically start at -24.57. While this might not always make sense practically, especially if the context doesn't align with negative values, mathematically it helps in forming the full equation of our line: \( y = -24.57 + 15.75x \). This completes the picture on how the line fits the given dataset.

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Most popular questions from this chapter

Find the equation of the least-squares line for the given data. Graph the line and data points on the same graph. The pressure \(p\) was measured along an oil pipeline at different distances from a reference point, with results as shown. Find the least-squares line for \(p\) as a function of \(x\) using a calculator. Then predict the pressure at a distance of \(x=500 \mathrm{ft}\). Is this interpolation or extrapolation? $$\begin{array}{l|r|l|l|l|l} x(\mathrm{ft}) & 0 & 100 & 200 & 300 & 400 \\ \hline p\left(\mathrm{lb} / \mathrm{in.}^{2}\right) & 650 & 630 & 605 & 590 & 570 \end{array}$$

Find the equation of the least-squares line for the given data. Graph the line and data points on the same graph. If gas is cooled under conditions of constant volume, it is noted that the pressure falls nearly proportionally as the temperature. If this were to happen until there was no pressure, the theoretical temperature for this case is referred to as absolute zero. In an elementary experiment, the following data were found for pressure and temperature under constant volume. $$\begin{array}{l|c|c|c|c|c|c} T\left(^{\circ} \mathrm{C}\right) & 0.0 & 20 & 40 & 60 & 80 & 100 \\ \hline P(\mathrm{kPa}) & 133 & 143 & 153 & 162 & 172 & 183 \end{array}$$ Use a calculator to find the least-squares line for \(P\) as a function of \(T,\) and from the graph determine the value of absolute zero found in this experiment.

Use the following data. In a random sample. 30 Android users were asked to record the number of apps that were installed on their phone. The resulting data are shown below: $$\begin{aligned} &112,91,101,85,76,115,93,126,78,86,105,107,58,86,109\\\ &111,103,105,97,110,92,95,107,89,101,67,103,99,93,82 \end{aligned}$$ Make a stem-and-leaf plot of these data using split stems.

Use the following information. If the weights of cement bags are normally distributed with a mean of 60 lb and a standard deviation of 1 lb, use the empirical rule to find the percent of the bags that weigh the following: Less than \(63 ~ \mathrm{lb}\)

Use the following information. A standardized math test has a mean score of 200 and a standard deviation of \(15 .\) Find and interpret the z-scores of the following math test scores. $$233$$
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