Using the normal distribution, we have that the desired information is given as follows:
a) X ~ N(3,1.8).
b) 3 days.
c) Z = 0.5556.
d) 0.33 = 33%.
e) 0.2048 = 20.48%.
f) 5.3 days.
<h3>Normal Probability Distribution</h3>
The z-score of a measure X of a normally distributed variable with mean and standard deviation is given by:
- The z-score measures how many standard deviations the measure is above or below the mean.
- Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.
The mean and the standard deviation are given, respectively, by:
Hence the distribution is X ~ N(3,1.8), and, as is standard in the normal distribution, the median is the same as the mean, which is of 3 days.
The Z-score for a patient that took 4 days to recover is <u>Z when X = 4</u>, hence:
Z = (4 - 3)/1.8 = 0.5556.
The probability of spending more than 3.8 days in recovery is <u>one subtracted by the p-value of Z when X = 3.8</u>, hence:
Z = (3.8 - 3)/1.8 = 0.44.
Z = 0.44 has a p-value of 0.67.
1 - 0.67 = 0.33.
The probability is of 0.33 = 33%.
The probability of spending between 3.2 and 4.2 days in recovery is the <u>p-value of Z when X = 4.2 subtracted by the p-value of Z when X = 3.2</u>, hence:
X = 4.2:
Z = (4.2 - 3)/1.8
Z = 0.67.
Z = 0.67 has a p-value of 0.7486.
X = 3.2:
Z = (3.2 - 3)/1.8
Z = 0.11.
Z = 0.11 has a p-value of 0.5438.
0.7486 - 0.5438 = 0.2048 = 20.48%.
The 90th percentile is <u>X when Z = 1.28,</u> hence:
1.28 = (X - 3)/1.8
X - 3 = 1.8 x 1.28
X = 5.3 days.
More can be learned about the normal distribution at brainly.com/question/15181104
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