Answer: the 90th percentile for recovery times is 8.77 days.
Explanation:
Let x be the random variable representing the recovery time of patients from a particular surgical procedure. Since it is normally distributed and the population mean and population standard deviation are known, we would apply the formula,
z = (x - µ)/σ
Where
x = sample mean
µ = population mean
σ = standard deviation
From the information given,
µ = 5.7 days
σ = 2.4 days
The probability for the 90th percentile is 90/100 = 0.9
The z score corresponding to the probability value on the normal distribution table is 1.28
Therefore,
1.28 = (x - 5.7)/2.4
Cross multiplying, it becomes
1.28 × 2.4 = x - 5.7
3.072 = x - 5.7
x = 3.072 + 5.7 = 8.77 days