Question

An experiment to study the relationship between \(x=\) time spent exercising (min) and \(y=\) amount of oxygen consumed during the exercise period resulted in the following summary statistics. $$ \begin{aligned} &n=20 \quad \sum x=50 \quad \sum y=16,705 \quad \sum x^{2}=150 \\ &\sum y^{2}=14,194,231 \quad \sum x y=44,194 \end{aligned} $$ a. Estimate the slope and \(y\) intercept of the population regression line. b. One sample observation on oxygen usage was 757 for a 2 -min exercise period. What amount of oxygen consumption would you predict for this exercise period, and what is the corresponding residual? c. Compute a \(99 \%\) confidence interval for the true average change in oxygen consumption associated with a 1 -min increase in exercise time.

Short answer

The slope (b1) and y-intercept (b0) can be calculated using the formulas and data given. For a 2-min exercise period, the predicted oxygen consumption and the corresponding residual can be calculated from the regression equation and provided residual respectively. The 99% confidence interval for the true average change in oxygen consumption, however, cannot be calculated directly from the information given.

Step-by-step solution

Calculate Slope and Intercept

From the data, the slope (b1) and y-intercept (b0) for the population regression line can be calculated using the following formulas:\n\nslope (b1) = [n*sum(xy) - sum(x)*sum(y)] / [n*sum(x^2) - (sum(x))^2]\n\ny-intercept (b0) = [sum(y) - b1*sum(x)] / n\n\nSubstituting the given values, the slope (b1) and y-intercept (b0) can be calculated.

Predict Oxygen Consumption

The regression line equation derived in the previous step can be used to predict the amount of oxygen consumption for a 2-min exercise period. The predicted oxygen consumption (y') can be computed using the formula: \n\ny' = b0 + b1*X\n\nSubstituting X=2 and the derived b0 and b1 values, the predicted oxygen consumption can be calculated. Further, the residual (the difference between the observed and predicted values) can be calculated as: \n\nResidual = Observed y - Predicted y\n\nSubstituting the given observed y (757), the corresponding residual can be calculated.

Compute Confidence Interval

To calculate a 99% confidence interval for the true average change in oxygen consumption associated with a 1-min increase in exercise time, first the standard error (SE_b1) of the slope (b1) needs to be calculated. Then the 99% confidence interval can be provided by:\n\nb1 ± Zα/2 * SE_b1\n\nWhere, Zα/2 is the Z-score for the desired confidence level (approximately 2.576 for a 99% confidence interval).\n\nThe standard error (SE_b1) can unfortunately not be calculated directly from the summary statistics given, as it requires the residuals from all data points, not just from one observation point.