The grades of a class of 9 students on a midterm report (x)...

The grades of a class of 9 students on a midterm report $(x)$ and on the final examination $(y)$ are as follows: $$ \begin{array}{c|ccccccccc} x & 77 & 50 & 71 & 72 & 81 & 94 & 96 & 99 & 67 \\ \hline y & 82 & 66 & 78 & 34 & 47 & 85 & 99 & 99 & 68 \end{array} $$ (a) Estimate the linear regression line. (b) Estimate the final examination grade of a student who received a grade of 85 on the midterm report.

The grades of a class of 9 students on a midterm report $(x)$ and on the final examination $(y)$ are as follows:
$$
\begin{array}{c|ccccccccc}
x & 77 & 50 & 71 & 72 & 81 & 94 & 96 & 99 & 67 \\
\hline y & 82 & 66 & 78 & 34 & 47 & 85 & 99 & 99 & 68
\end{array}
$$
(a) Estimate the linear regression line.
(b) Estimate the final examination grade of a student who received a grade of 85 on the midterm report.

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Key Concepts

Linear Regression

Linear regression is a fundamental statistical method used to model the relationship between two quantitative variables by fitting a linear equation to observed data. This method allows us to explain how the dependent variable changes as the independent variable changes, and it serves as a basic predictive tool by identifying trends in the data.

Least Squares Method

The least squares method is a standard approach for estimating the parameters of a linear regression model. It does so by minimizing the sum of the squares of the differences (residuals) between the observed values and the values predicted by the model. This technique ensures that the best-fitting line minimizes the overall error in prediction.

Prediction Using Regression

Prediction using regression involves using the estimated linear equation to estimate the value of the dependent variable for a given value of the independent variable. Once the regression model has been computed, it can be employed to predict outcomes, making it a powerful tool for forecasting future results based on historical data.

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