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The t-statistic is an essential part of determining the significance of the slope in a linear regression model. It is primarily used to test the hypothesis that the slope (\( \beta_1 \)) is zero. Essentially, it helps us understand whether there is a meaningful linear relationship between our independent and dependent variables. The formula for the t-statistic is given by \[ t = \frac{\hat{\beta_1}}{\text{SE}(\hat{\beta_1})} \]where \( \hat{\beta_1} \) is the estimated slope, and \( \text{SE}(\hat{\beta_1}) \) is the standard error of this estimate.The key idea is that if the t-statistic is far from zero, it suggests that the slope is significantly different from zero, implying a strong linear relationship. To conduct the hypothesis test, you'd compare the computed t-value to a critical value from a t-distribution table based on your data's degrees of freedom. This will tell you whether to reject the null hypothesis (no linear relationship).

In the context of linear regression, slope estimation is the process of determining the line's steepness in relation to the independent variable, represented by \( \hat{\beta_1} \). This slope defines the change in the dependent variable for a one-unit increase in the independent variable. The formula for estimating the slope is:\[\hat{\beta_1} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\]Here, \( x_i \) and \( y_i \) are the individual data points; \( \bar{x} \) and \( \bar{y} \) are the means of the independent and dependent variables, respectively. This formula essentially calculates how much the two variables move together, contrasting it with the variability in the independent variable alone.When you scale the variables, the slope adjusts proportionately if you multiply these variables by constants, as shown in the exercise. The crux here is that scaling both variables by respective constants still maintains the underlying relationship, and thus does not alter the t-statistic value.

The standard error (\( \text{SE}(\hat{\beta_1}) \)) is a measure that tells us the average amount that the slope estimate \( \hat{\beta_1} \) might deviate from the actual value of \( \beta_1 \). This forms the basis for the variability and precision of our estimate.The formula for the standard error of the slope is:\[\text{SE}(\hat{\beta_1}) = \sqrt{\frac{\sigma^2}{\sum (x_i - \bar{x})^2}}\]where \( \sigma^2 \) is the error variance, reflecting the dispersion of data points around the fitted regression line. When scaling occurs, as detailed in the solution provided, the standard error also scales proportionately with changes in the data. However, in the context of this exercise, since both the numerator and the denominator in the t-statistic formula scale similarly, the t-statistic itself does not change. The standard error is crucial as it affects the hypothesis testing process. A smaller standard error indicates a more precise estimate of the slope, making your hypothesis tests more reliable.