Question

Show some computer output for fitting simple linear models. State the value of the sample slope for each model and give the null and alternative hypotheses for testing if the slope in the population is different from zero. Identify the p-value and use it (and a \(5 \%\) significance level) to make a clear conclusion about the effectiveness of the model.$$ \begin{array}{lrrrr} \text { Coefficients: } & \text { Estimate } & \text { Std.Error } & \mathrm{t} \text { value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \text { (Intercept) } & 7.277 & 1.167 & 6.24 & 0.000 \\ \text { Dose } & -0.3560 & 0.2007 & -1.77 & 0.087 \end{array} $$

Short answer

The sample slope for 'Dose' is -0.3560. The null hypothesis stating that the population slope equals 0 cannot be rejected at a 5% significance level as the p-value (0.087) is greater than 0.05. Therefore, there is not enough evidence to suggest that 'Dose' has a significant effect.

Step-by-step solution

Understanding the model results

Based on the given computer output, the sample slope (also known as the coefficient for the predictor variable) for the 'Dose' variable is -0.3560. This means that for each unit increase in 'Dose', our outcome variable is predicted to decrease by approximately 0.356, assuming all other variables remain constant.

State the null and alternative hypotheses

For the 'Dose' variable, the null (H0) and alternative (H1) hypotheses for testing if the slope in the population is different from zero are: H0: The population slope = 0 (There is no effect of 'Dose' on the outcome variable.)H1: The population slope ≠ 0 (There is an effect of 'Dose' on the outcome variable.)

Identification of p-value

The p-value for 'Dose' from the output is 0.087. The p-value signifies the probability that you would obtain the observed statistic if the null hypothesis were true.

Conclusion about the model's effectiveness

If a given p-value is less than the significance level (0.05 in this case), we reject the null hypothesis. Here, the p-value (0.087) is greater than the 0.05 significance level. Therefore, we do not reject the null hypothesis and conclude that there is not enough evidence to suggest the 'Dose' has a significant effect at a 5% significance level.