Question

Life Expectancy In Exercise 9.27 on page 607 , we consider a regression equation to predict life expectancy from percent of government expenditure on health care, using data for a sample of 50 countries in SampCountries. Using technology and this dataset, find and interpret a \(95 \%\) prediction interval for each of the following situations: (a) A country which puts only \(3 \%\) of its expenditure into health care. (b) A country which puts \(10 \%\) of its expenditure into health care. (c) A country which puts \(50 \%\) of its expenditure into health care. (d) Calculate the widths of the intervals from (a), (b), and (c). What do you notice about these widths? (Note that for this sample, government expenditures on health care go from a minimum of \(4.0 \%\) to a maximum of \(20.89 \%\), with a mean of \(12.31 \% .)\)

Short answer

After calculating the 95% prediction intervals, the intervals for a country putting 3%, 10% and 50% of its expenditure into health care are established. The widths of these intervals are then found by subtracting the lower limit from the upper limit. It can be observed that the prediction interval width is narrowest for the country that invests closest to the mean health care expenditure of the sample (12.31%) and widest for countries putting either 3% or 50% of its expenditure into health care. The exact numbers would be based on the parameters of the given dataset.

Step-by-step solution

Establish the prediction interval formula

Given a regression model, the prediction interval for an individual response \(y_i\) at \(x_i\) is given by \n\(y_i = b_0 + b_1 x_i ± t_{α/2,n-2} s_e \sqrt{1 + 1/n + (x_i - x̄)^2 / ∑(x-x̄)^2}\)\nThis formula is used when we are trying to predict Y for a given value of X. The variable \(b_0\) is the y-intercept of the regression line, \(b_1\) is the slope, \(x_i\) is the given X value, \(t_{α/2,n-2}\) is the t-value from the t-distribution table where α is the confidence level and n is the number of observations, \(s_e\) is the standard error of estimate, n is the total number of observations, \(x̄\) is the mean of X and \(∑(x-x̄)^2\) is the sum of the squares of the differences between each X value and the mean of X.

Calculation for a country putting 3% of its expenditure into health care

Given that only 3% of expenditure goes into health care for a particular country, this means \(x_i = 3\). We substitute this value into the prediction interval formula mentioned in Step 1 with an established regression line and parameters of the dataset. We perform the calculation to find the lower and upper limits of the prediction interval.

Calculation for a country putting 10% of its expenditure into health care

Given that 10% of expenditure goes into health care, substituting \(x_i = 10\) into the prediction interval formula. Repeat the process as in Step 2 to find the prediction interval.

Calculation for a country investing 50% of its expenditure into health care

Considering that 50% of expenditure goes into health care making \(x_i = 50\), we substitute it in the prediction interval formula. Though, it is important to note that this situation may not provide an accurate prediction since 50% falls way outside the range of health care expenditure percentages in the provided dataset. Generally, the regression model is best for predicting within the range of the available data.

Calculate the widths of the intervals

The width of the prediction interval is the difference between the upper and lower limits of the interval. It can be calculated for all scenarios - 3%, 10%, and 50% investment in health care - by simply subtracting the lower limit from the upper limit.

Analyze the widths of the intervals

The prediction interval width would typically narrow as the value of X gets closer to the mean of X. This is due to less uncertainty when the prediction is closer to where most of the data is located. Therefore, the interval would likely be narrowest for a country putting around 12.31% (the mean) of its expenditures into health care and wider for countries putting either 3% or 50% of its expenditure into health care.