Question

Survival in the ICU and Infection In the dataset ICUAdmissions, the variable Status indicates whether the ICU (Intensive Care Unit) patient lived (0) or died \((1),\) while the variable Infection indicates whether the patient had an infection ( 1 for yes, 0 for no) at the time of admission to the ICU. Use technology to find a \(95 \%\) confidence interval for the difference in the proportion who die between those with an infection and those without.

Short answer

The 95% confidence interval for the difference in the proportion of deaths between those with an infection and those without, is given by \([d - 1.96*SE, d + 1.96*SE]\), where \(d\) is the difference in sample proportions and \(SE\) is the standard error.

Step-by-step solution

Identify Sample Sizes and Proportions

First, from the dataset 'ICUAdmissions', identify the number of people in each group (with infection and without infection) and the number of deaths in each group. These will be your sample sizes and observed deaths.

Calculate Sample Proportions

Next, calculate the proportion of deaths in each group by dividing the number of deaths by the total number in each group. Let's denote the proportion of deaths in infected group as \(P1\) and in the non-infected group as \(P2\). Then find the difference in sample proportions, \(d = P1 - P2\).

Compute Standard Error

The standard error \((SE)\) for the difference in sample proportions is calculated using the formula \(SE = \sqrt{[P1*(1 - P1) / n1] + [P2*(1 - P2) / n2]}\), where \(n1\) and \(n2\) are the sample sizes of the infected and non-infected groups respectively.

Calculate Confidence Interval

The formula for a 95% confidence interval for the difference in proportions is \([d - 1.96*SE, d + 1.96*SE]\). Substitute the values of \(d\) and \(SE\) in the formula to get the 95% confidence interval for the difference in the proportions of deaths between the group with infection and the group without infection.