Final answer:
Rejecting the null hypothesis implies there is sufficient evidence to suggest the associated predictor variable influences the response variable significantly, but it does not prove the alternative hypothesis to be true absolutely.
Step-by-step explanation:
In hypothesis testing, when the null hypothesis (H0: βj = 0) is rejected in favor of the alternative hypothesis (HA: βj ≠ 0), it implies that there is sufficient statistical evidence to suggest that the population parameter βj (e.g., a slope in regression analysis) is significantly different from zero. Rejecting the null hypothesis suggests that the predictor variable associated with βj has a statistically significant relationship with the response variable. It is important to remember that this does not prove the alternative hypothesis true; it just indicates that the sample data is not consistent with the null hypothesis.
To determine whether to reject or not reject the null hypothesis, the p-value is compared to a predefined significance level, denoted by α. If the p-value is less than α, the null hypothesis is rejected; otherwise, it is not rejected. The decision to reject H0 does not mean HA is true beyond any doubt; it means that the evidence against H0 in the context of the test is strong enough to favor HA.
Performing a hypothesis test involves specifying the types of errors that can occur. Rejecting H0 when it is true is known as a Type I error, while failing to reject H0 when it is false is known as a Type II error. The probability of making a Type I error is α, which is the significance level of the test. When analyzing the outcome of a hypothesis test, one must always be conscious of these errors and the limitations they impose on the conclusions that can be drawn.