The following table shows students' test scores on the first two tests in an introductory chemistry class.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline \multicolumn{10}{|c|}{ Chemistry Test Scores } \\
\hline \begin{tabular}{c}
First \\
test, $\boldsymbol{x}$
\end{tabular} & 86 & 82 & 47 & 69 & 73 & 47 & 63 & 50 & 59 & 77 & 88 & 87 \\
\hline \begin{tabular}{c}
Second \\
test, $\boldsymbol{y}$
\end{tabular} & 81 & 71 & 51 & 75 & 69 & 44 & 48 & 48 & 58 & 68 & 83 & 77 \\
\hline
\end{tabular}

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Step 1 of 2 : Find an equation of the least-squares regression line. Round your answer to three decimal places, if necessary.

Answer
Tables

Step 1 ：We are given the scores of students on the first and second tests in an introductory chemistry class. The scores for the first test are represented by x and the scores for the second test are represented by y. The scores are as follows: x = [86, 82, 47, 69, 73, 47, 63, 50, 59, 77, 88, 87] and y = [81, 71, 51, 75, 69, 44, 48, 48, 58, 68, 83, 77].

Step 2 ：We are asked to find the equation of the least-squares regression line for the given data. The least-squares regression line is a line that minimizes the sum of the squares of the vertical deviations from each data point to the line. The equation of the least-squares regression line is given by \(y = mx + b\), where m is the slope and b is the y-intercept.

Step 3 ：The slope m is given by the formula \(m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}\), and the y-intercept b is given by the formula \(b = \frac{\Sigma y - m(\Sigma x)}{n}\), where n is the number of data points, \(\Sigma xy\) is the sum of the product of each pair of x and y values, \(\Sigma x\) and \(\Sigma y\) are the sums of the x and y values respectively, and \(\Sigma x^2\) is the sum of the squares of the x values.

Step 4 ：Using the given data, we find that n = 12, \(\Sigma x\) = 828, \(\Sigma y\) = 773, \(\Sigma xy\) = 55550, and \(\Sigma x^2\) = 59820.

Step 5 ：Substituting these values into the formulas, we find that the slope m is approximately 0.823 and the y-intercept b is approximately 7.61.

Step 6 ：Substituting the slope and y-intercept into the equation \(y = mx + b\), we get the equation of the least-squares regression line.

Step 7 ：Final Answer: The equation of the least-squares regression line is \(\boxed{y = 0.823x + 7.61}\).