Question

3.

a. A researcher wants to study how long (in days) patients aged 30-39 who are admitted to a hospital for coronavirus treatment spend in the hospital before being discharged. Preliminary data suggests that
? = 4 days is a safe assumption. Assuming the recovery times have standard deviation 4 days (which would say that virtually all patients recover in some number of days +/- about 12 days), use the formula

? = ?^2 * ?^2 / ?^2

To find the number of patients that would need to be sampled to construct a 95% confidence interval for the mean recovery time with a margin of error of 0.5 days. Make sure to show how you find the critical value z

b. Same question as part a, if instead you assume that ? = 3.5 days.

Answer 1

Solution(a)
Given in the question
Population standard deviation() = 4
Margin of error(E) = 0.5 days
Here we will use 95% confidence interval to calculate sample size
So alpha = 1 - 0.95 = 0.05
alpha/2 = 0.05/2 = 0.025
From Z-table we found Zalpha/2 as shown in figure = 1.96
So Z-value = 1.96
Sample size can be calculated as
n = ?^2 * ?^2 / ?^2 = (1.96*1.96*4*4)/(0.5*0.5) = 61.4656/0.25 = 245.86 or 246
Solution(b)
Here Population standard deviation() = 3.5
and rest all values are same so
Sample size can be calculated as
n = ?^2 * ?^2 / ?^2 = (1.96*1.96*3.5*3.5)/(0.5*0.5) = 47.0596/0.25 = 188.24 or 188