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The least squares regression method helps us find a straight line that best represents the data. It uses experience as the independent variable and monthly salary as the dependent variable. This means we predict salary based on experience. To do this, we calculate two important components: the slope (B1) and the intercept (B0).
We start with calculating the slope, which shows how much the salary changes with each year of experience. The formula for the slope B1 is:

• \(B1 = \frac{\Sigma((x_i - \overline{x})(y_i - \overline{y}))}{\Sigma(x_i - \overline{x})^2}\)

Here, \(\overline{x}\) is the mean experience, and \(\overline{y}\) is the mean salary. Once we find the slope, we can find the intercept B0 using:

• \(B0 = \overline{y} - B1*\overline{x}\)

Put together, the regression line equation is:

• \(y = B0 + B1*x\)

This equation allows us to predict salaries from given years of experience.

Confidence intervals provide a range of values within which we can reasonably expect to find the true parameter. In regression, we often calculate the confidence interval for the slope (B1), showing the possible range of the effect of experience on salary.
For the 98% confidence interval, the formula is:

• \(B1 \pm t*SE(B1)\)

The \(t\) value is determined by the degrees of freedom and the desired confidence level, found in t-distribution tables. The \(SE(B1)\) represents the standard error of B1.
This interval estimates the range where the true slope value should lie 98% of the time. If the interval does not include zero, it's evidence that a linear relationship exists.

Hypothesis testing in regression helps determine if our discovered relationships are statistically significant. For the slope (B1), the null hypothesis usually states that B1 equals zero, implying no relationship between experience and salary.
To test this, we use the formula:

• \(T = \frac{B1-0}{SE(B1)}\)

Where \(T\) is the test statistic, compared against a critical value from the t-distribution for the desired confidence level. If the \(T\) statistic exceeds this critical value at a 2.5% significance level, we reject the null hypothesis. This suggests a significant positive relationship exists.

Understanding independent and dependent variables is fundamental in regression analysis. In our exercise, experience is the independent variable, while the salary is the dependent variable.
- **Independent Variable:** This is the factor we change or control to observe effects on dependent variables. - **Dependent Variable:** This variable changes in response to the independent variable. It's what you measure in the experiment.
By manipulating the years of experience, we observe how it affects the salary, making this model predictive. Such relationships help understand causalities within data, predicting changes in the dependent variable based on known changes in the independent variable.