Question

Regression equation Studies have shown that people who suffer sudden cardiac arrest have a better chance of survival if a defibrillator shock is administered very soon after cardiac arrest. How is the survival rate related to the time between when cardiac arrest occurs and when the defibrillator shock is delivered? This question is addressed in the paper "Improving Survival from Sudden Cardiac Arrest: The Role of Home Defibrillators" (by J.K Stross, University of Michigan, February 2002) The following statistics are provided: Mean call-to-shock time (x) xˉsx=7.2=3.701 ​Survival Rate pct (y) yˉ=34. s y =35.837 ​The correlation coefficient is r=−0.960 Using these values, - Calculate the regression equation - How much will the survival rate change for each minute that a defibrillator is delayed being administered? Recall: y^ =bx+ab=r s xs ya= yˉ −b xˉ

Answer 1

The regression equation for the given data is
`\( \hat{y} = 60.12 - 2.93x \)`. For each minute that a defibrillator is delayed, the survival rate is expected to decrease by approximately 2.93%.

Step-by-step explanation:

The regression equation is derived using the given correlation coefficient (r), mean call-to-shock time x, and survival rate y. The formula for the regression equation is
`\( \hat{y} = a + bx \)`, where \(a\) is the y-intercept and(b is the slope. Substituting the provided values, the equation is calculated as
`\( \hat{y} = 60.12 - 2.93x \)`.

To understand the impact of delay, we look at the coefficient of \(x\) in the equation. In this case, for each minute that a defibrillator is delayed (\x increases by 1), the predicted survival rate
`(\(\hat{y}\))` is expected to decrease by approximately 2.93%. This information is crucial for assessing the criticality of prompt defibrillation in improving the chances of survival after sudden cardiac arrest.

In medical contexts, such regression analyses help quantify the relationship between variables, providing valuable insights for decision-making and intervention strategies. The negative slope in this case emphasizes the importance of minimizing the time between cardiac arrest and defibrillator shock to enhance survival rates.