The patient recovery time from a particular surgical procedure is normally distributed with a mean of 3 days and a standard deviation of 1.6 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? days c. What is the Z-score for a patient that took 4.2 days to recover? d. What is the probability of spending more than 3.8 days in recovery? e. What is the probability of spending between 2.3 and 2.8 days in recovery? f. The 70th percentile for recovery times is? days. The number of ants per acre in the forest is normally distributed with mean 42,000 and standard deviation 12,221. Let X = number of ants in a randomly selected acre of the forest. Round all answers to 4 decimal places where possible. a. What is the distribution of X? X ~ N(. , ) b. Find the probability that a randomly selected acre in the forest has fewer than 32,793 ants? c. Find the probability that a randomly selected acre has between 46,676 and 51,692 ants? d. Find the first quartile? ants (round your answer to a whole number)
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
The patient recovery time from a particular surgical procedure is
a. What is the distribution of X? X ~ N( , )
b. What is the median recovery time? days
c. What is the Z-score for a patient that took 4.2 days to recover?
d. What is the probability of spending more than 3.8 days in recovery?
e. What is the probability of spending between 2.3 and 2.8 days in recovery?
f. The 70th percentile for recovery times is? days.
The number of ants per acre in the forest is normally distributed with mean 42,000 and standard deviation 12,221. Let X = number of ants in a randomly selected acre of the forest. Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(. , )
b. Find the probability that a randomly selected acre in the forest has fewer than 32,793 ants?
c. Find the probability that a randomly selected acre has between 46,676 and 51,692 ants?
d. Find the first quartile? ants (round your answer to a whole number)
Solution:
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Let X be the recovery time for a randomly selected patient.
From the given information, X follows normal distribution with mean µ=3 days and a standard deviation σ=1.6 days.
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