Question

In Exercise 9.27 we see that the conditions are met for fitting a linear model to predict life expectancy (LifeExpectancy) from the percentage of government expenditure spent on health care (Health) using the data in SampCountries. Use technology to examine this relationship further, as requested below. (a) Find the correlation between the two variables and give the p-value for a test of the correlation. (b) Find the regression line and give the t-statistic and p-value for testing the slope of the regression line. (c) Find the F-statistic and the p-value from an ANOVA test for the effectiveness of the model. (d) Comment on the effectiveness of this model.

Short answer

This short answer section will be filled after performing the actual numerical calculations. Hence it will be based on the findings from the statistical software analysis.

Step-by-step solution

Calculate the correlation

First, you must calculate the correlation between the 'LifeExpectancy' and 'Health' variables using a statistical software. The correlation will give you an idea of the strength and direction of the linear relationship between these two variables. Also, perform a hypothesis test to find out the p-value. The null hypothesis is that the correlation is zero, and the alternative is that it is not zero.

Calculate the regression line

Then, calculate the regression line to predict 'LifeExpectancy' based on 'Health'. The equation will be of the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the intercept. Again, use your statistical software to perform a hypothesis test for the slope. The null hypothesis is that the slope is zero (indicating no relationship between 'LifeExpectancy' and 'Health'), and the alternative is that it is not zero.

Perform an ANOVA test

Next, perform an analysis of variance (ANOVA) test. This will calculate the F-statistic, which tests the overall significance of the model. As with the previous steps, use statistical software to find the F-statistic and p-value.

Comment on the model's effectiveness

Finally, based on your analyses in the earlier steps, provide a comment on the model's effectiveness. If the p-values from your hypothesis tests are less than your chosen significance level (typically 0.05), then the model can be considered effective. Additionally, consider the practical implications of your findings when making this comment.