Slopes

Performing multiple linear regressions including other lifestyle factors such as diet and sleeping patterns

Calculating Pearson’s correlation coefficient between reported exercise frequency and stress levels from surveys

Using historical data on exercise routines and medical records documenting stress-related conditions

Conducting a randomized controlled trial measuring stress responses after controlled exercise interventions.

$H_{0} : \beta_{1} = 0$

$H_{0} : \beta_{1} > 0$

$H_{0} : \mu = \mu_{0}$ where $\mu$ is population mean score

$H_{0} : r = 0$ where $r$ is Pearson's correlation coefficient

Accept the null hypothesis

Reject the null hypothesis

Cannot determine the conclusion

Fail to reject the null hypothesis

Proportion of variance in Y explained by X.

Change in response variable per one unit increase in explanatory variable.

Average value of response variable Y when explanatory variable X equals zero.

Difference between actual and predicted values of Y.

There is sufficient evidence to reject the null hypothesis that the slope is zero.

The correlation between variables in the model is not significant.

A type II error has definitely been made in testing the hypothesis.

There is not enough evidence to reject the null hypothesis that the slope is zero.

Cannot determine the conclusion

Reject the null hypothesis

Fail to reject the null hypothesis

Accept the null hypothesis

Uniform distribution

Exponential distribution

t-distribution

Binomial distribution

Decreases the degrees of freedom

Increases the significance level

Increases the likelihood of rejecting the null hypothesis

Increases the likelihood of failing to reject the null hypothesis

Homoscedastic distribution of residuals as anticipated under correct modelling assumptions.

Residuals evenly dispersed around both sides of zero without any discernible trend.

Residuals displaying random variability without any clusters indicative optimal conditions.

A clear curvature or pattern within residual plots contrary to randomness expected under adequacy.

Cannot determine the conclusion

Reject the null hypothesis

Fail to reject the null hypothesis

Accept the null hypothesis