Hypothesis Test and Confidence Interval for Regression Slope

Hypothesis Test for Slope

1. Hypotheses:

Null Hypothesis (H0): The slope of the regression line is equal to zero (no linear relationship between 'cath_width' and 'cath_height').
Alternative Hypothesis (HA): The slope of the regression line is not equal to zero (there is a linear relationship between 'cath_width' and 'cath_height').

Test Statistics and Degrees of Freedom:

The test statistic is the t value, which is calculated as the coefficient estimate divided by its standard error. In this case, the t value for the slope ('cath_width') is 2.362e+16.
The degrees of freedom (df) are given as 98, which represents the number of observations minus the number of predictors (in this case, 2 - intercept and 'cath_width').

P-value:

Conclusion:

Since the p-value is less than the significance level (usually 0.05), we reject the null hypothesis.
There is sufficient evidence to conclude that there is a linear relationship between 'cath_width' and 'cath_height'.

2. 95% Confidence Interval for the Slope:

The slope coefficient estimate is 8.947e+00.
The standard error for the slope coefficient is 3.788e-16.
The 95% confidence interval for the slope can be calculated by taking the estimate ﾱ (critical value * standard error).
The critical value for a 95% confidence interval with 98 degrees of freedom can be obtained from the t-distribution table.

Explanation:

The 95% confidence interval for the slope represents the range of values within which we can be 95% confident that the true slope coefficient lies.
If the confidence interval contains zero, it suggests that the slope is not significantly different from zero and there may not be a linear relationship between the variables.
If the confidence interval does not contain zero, it indicates that the slope is significantly different from zero and there is evidence of a linear relationship between the variables.