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The Coefficient of Determination, denoted as \(r^2\), is a key concept in simple linear regression. It helps in understanding how well the data fits the regression model. In simpler terms, it tells us the proportion of variation in the dependent variable (sales revenue, in this case) that can be explained by the independent variable (advertising expenditure). An \(r^2\) value closer to 1 indicates a strong relationship, while a value closer to 0 suggests a weaker relationship.
To compute \(r^2\), use the formula:

• \(r^2 = 1 - \frac{ \sum (y-\hat{y})^2}{\sum (y-\bar{y})^2}\)

Here:- \(\sum (y-\hat{y})^2\) is the sum of the squares of the residuals, showing variation unexplained by the model.
- \(\sum (y-\bar{y})^2\) is the total variation in the data.Plugging in the given values, such as \(\sum(y-\hat{y})^{2}=561.46\) and \(\sum(y-\bar{y})^{2}=2401.85\), shows the extent to which advertising affects sales revenue by computing \(r^2\). This proportion helps businesses understand the effectiveness of advertising in influencing sales.

A Confidence Interval provides a range of values that is likely to contain the true parameter value of interest. In the context of regression analysis, it helps to estimate the true slope \(\beta\) of the linear relationship between the independent and dependent variables.
For a 90% confidence interval, you first determine the slope \(\hat{\beta}\) of the regression line through calculations not detailed here but using the given data. Next, use the t-distribution to find \(t_{\alpha/2}\), which accounts for the desired confidence level and sample size.
The formula for the confidence interval is:

• \(\hat{\beta} - (t_{\alpha/2} \times s_{b}) \leq \beta \leq \hat{\beta} + (t_{\alpha/2} \times s_{b})\)

Where:- \(s_{b}\) is known as the standard error of the slope, determined through specific calculations involving residual variances.
- \(t_{\alpha/2}\) is a critical value from the t-distribution table.This interval provides a range where the average change in sales revenue per $1000 change in advertising expenditure may lie, with 90% confidence.

Standard Error in regression analysis quantifies how much estimated regression coefficients might differ when computed from various random samples. It's crucial for assessing the precision of the estimated coefficients like the slope.
There are two types to consider:

• \(s_e\): The standard error of the estimate. It measures the accuracy of predictions made by the regression line, representing the average distance the observed values fall from the regression line.
  - \(s_e = \sqrt{\frac{\sum (y-\hat{y})^2}{n-2}}\), where \(n\) is the number of data points.

• \(s_b\): The standard error of the slope. It indicates how much the estimated slope \(\hat{\beta}\) can vary from the true slope \(\beta\).
  - Calculated as \(s_b = s_e \times \sqrt{\frac{1}{\sum (x-\bar{x})^2}}\), with \(\bar{x}\) as the mean of the \(x\) values.

Understanding these errors assures that the interpretations drawn from the regression model are within plausible reliability limits, enhancing inference accuracy in real-world applications.