Final answer:
The hypotheses for a t-test of the slope involve testing if the regression line's slope is significantly different from a hypothesized value, usually zero. The null hypothesis asserts no relationship, while the alternative suggests some effect. The t-score and p-value determine if the null hypothesis can be rejected.
Step-by-step explanation:
The hypotheses for a t-test of the slope involve comparing the estimated slope of a regression line to a hypothesized value, typically zero. In the case of the golf instructor assessing the effectiveness of a new technique, we are concerned with whether the slope of the regression line, representing the change in golf scores after taking the class, is significantly different from zero.
In a hypothesis test for the slope, the null hypothesis (H0) posits that there is no effect or no relationship, indicating that the true slope (β) is equal to zero. The alternative hypothesis (Ha) suggests that there is an effect, and therefore the slope is not equal to zero. It can be two-sided (β ≠ 0), left-sided (β < 0), or right-sided (β > 0) depending on the direction of the effect we are testing for.
In the application of a t-test, the t-score and corresponding p-value will indicate if there is sufficient evidence to reject the null hypothesis. A significantly low p-value (typically less than 0.05) suggests that the slope is significantly different from zero, meaning that the change in scores is statistically significant and could be attributed to the new technique.