Question

The patient recovery time from a particular surgical procedure is normally distributed with a mean of 3 days and a standard deviation of 1.5 days. Let X be the recovery time for a randomly selected patient. Round all answers to 4 decimal places where possible.a. What is the distribution of X? X ~ N(,)b. What is the median recovery time? daysc. What is the Z-score for a patient that took 4.7 days to recover? d. What is the probability of spending more than 2.3 days in recovery? e. What is the probability of spending between 4 and 4.7 days in recovery? f. The 75th percentile for recovery times is days.

Answer 1

The mean recovery time is 3 days.

The standard deviation is 1.5 days.

Part a:

Assuming a normal distribution, we have:

`\begin{gathered} X\text{\textasciitilde}N(\mu,\sigma^2) \\ X\text{\textasciitilde}N(3,2.25) \end{gathered}`

Part b:

Since it is a normal distribution, the median must be the same as the mean, that is 3 days.

Part c:

For a recovery time of 4.7 days, we have:

`\begin{gathered} Z=(4.7-\mu)/(\sigma) \\ Z=(4.7-3)/(1.5) \\ Z=1.1333 \end{gathered}`

Part d:

The Z-score for a recovery time of 2.3 days is given by:

`Z_(x=2.3)=(2.3-3)/(1.5)=-0.4667`

Then, according to a normal table, we have:

P(X > 2.3) = P(Z > -0.4667) = 0.6808

Part e:

The Z-score for 4 is given by:

`Z_(x=4)=(4-3)/(1.5)=0.6667`

Then we have:

[tex]P(4Part f:

According to the normal distributrion, the 75th percentile for recivery time is close to Z = 0.66, which corresponds to 4 days.