Final answer:
Testing for the existence of correlation in simple linear regression is equivalent to testing for the existence of the slope (β1). The correlation coefficient measures the strength and direction of the linear relationship between the independent variable (x) and the dependent variable (y).
Step-by-step explanation:
In simple linear regression, testing for the existence of correlation is equivalent to testing for the existence of the slope (β1). The correlation coefficient measures the strength and direction of the linear relationship between the independent variable (x) and the dependent variable (y). It ranges from -1 to +1, where a value of 0 indicates no correlation, a positive value indicates a positive correlation, and a negative value indicates a negative correlation.
To test for significance of the correlation coefficient, we use a hypothesis test to determine whether the correlation coefficient is significantly different from zero. If the p-value associated with the test is less than the chosen significance level (usually 0.05), we reject the null hypothesis and conclude that there is a significant linear relationship between x and y in the population.