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Use technology to find the regression line to predict \(Y\) from \(X\). $$\begin{array}{rrrrrrrrr} \hline X & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 \\\Y & 532 & 466 & 478 & 320 & 303 & 349 & 275 & 221 \\\\\hline\end{array}$$

Short Answer

Expert verified
To predict \(Y\) from \(X\), the regression line, which is described by the equation \(Y = a + bX\), can be used. Here, \(a\) is the y-intercept and \(b\) is the slope of the line. The specific values for \(a\) and \(b\) are to be calculated based on the provided data set.

Step by step solution

01

Understand what is required.

A regression line is a straight line that describes how a response variable \(y\) changes as an explanatory variable \(x\) changes. We want to generate this line for the provided data.
02

Find the means of \(X\) and \(Y\).

The means \(\bar{X}\) and \(\bar{Y}\) are the average values of \(X\) and \(Y\) respectively. We can find these by summing up all the values in each set and then dividing by the number of values. The means will be used in calculating the slope and y-intercept.
03

Calculate the slope of the regression line.

The slope of the line, \(b\), is given by the formula: \[b = \frac{ \sum (X - \bar{X})(Y - \bar{Y})}{ \sum (X - \bar{X})^2}\] This calculation gives us the slope which tells us the direction in which the line ascends or descends
04

Calculate the y-intercept of the regression line.

The y-intercept, \(a\), which is the point where the line crosses the y axis at \(X=0\), is calculated using the formula: \[a = \bar{Y} - b \bar{X}\] This gives us the y-intercept that will help us plot the regression line.
05

Formulate the regression line equation.

The regression line can then be formulated using the equation: \[Y = a + bX\]. This equation allows us to predict any value of \(Y\) from any value of \(X\) within our data set.

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Most popular questions from this chapter

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