a) 42.17% of healthy people will have CBF readings between 60 and 80.
b) The probability that one healthy person taken at random will have a CBF reading of less than 54 is approximately 0.0062.
c) The probability that a sample of 25 healthy persons taken at random will have a sample mean of CBF readings of less than 54 is approximately 0.00000062.
d) 0.13% of healthy people will mistakenly be diagnosed as at risk for a stroke.
a) To determine the proportion of healthy people with CBF readings between 60 and 80, we need to calculate the area under the standard normal curve between 0.75 (-1.75 standard deviations below the mean) and 0.375 (2.375 standard deviations above the mean).
Using a standard normal table or calculator, we find that this area is approximately 0.4217. Therefore, 42.17% of healthy people will have CBF readings between 60 and 80.
b) To determine the probability that one healthy person taken at random will have a CBF reading of less than 54, we need to calculate the area under the standard normal curve to the left of -2.5 (1.5 standard deviations below the mean).
Using a standard normal table or calculator, we find that this area is approximately 0.0062. Therefore, the probability that one healthy person taken at random will have a CBF reading of less than 54 is approximately 0.0062.
c) To determine the probability that a sample of 25 healthy persons taken at random will have a sample mean of CBF readings of less than 54, we need to consider the sampling distribution of the mean.
The sampling distribution of the mean is normally distributed with a mean equal to the population mean (74) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (16/√25 = 4).
Therefore, we need to calculate the area under the standard normal curve to the left of (54 - 74)/4 = -5.0. Using a standard normal table or calculator, we find that this area is approximately 0.00000062.
Therefore, the probability that a sample of 25 healthy persons taken at random will have a sample mean of CBF readings of less than 54 is approximately 0.00000062.
d) To determine the proportion of healthy people who will mistakenly be diagnosed as at risk for a stroke, we need to calculate the area under the standard normal curve to the left of -2.125 (2.875 standard deviations below the mean).
Using a standard normal table or calculator, we find that this area is approximately 0.0013.
Therefore, 0.13% of healthy people will mistakenly be diagnosed as at risk for a stroke.
Question
Cerebral blood flow (CBF) in the brains of healthy people is normally distributed with a mean of 74 and a standard deviation of 16. a) What proportion of healthy people will have CBF readings between 60 and 80? b) What is the probability that one healthy person taken at random will have a CBF reading of less than 54? c) What is the probability that a sample of 25 healthy persons taken at random will have a sample mean of CBF readings of less than 54? d) If a healthy person has a CBF reading below 40, he is classified at risk for a stroke. What proportion of healthy people will mistakenly be diagnosed as