A person’s blood glucose level and diabetesare closely related. Let x be a
random variable measured in milligrams of glucose per deciliter (1/10 of a liter)
of blood. After a 12-hour fast, the random variable x will have a distribution
that is approximately normal with mean μ=85 and standard deviation σ=25.
After 50 years of age, both the mean and the standard deviation tend to increase.
What is the probability that, for an adult (under 50 years old) after 12-hour
fast,
(a) x is more than 60?
(b) x is less than 110?
(c) x is between 60 and 110?
(d) x is greater than 140 (borderline diabetes starts at 140)?
Z = (score - mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores found.
To find the probabilities in this scenario, we can use the normal distribution and the properties of the standard normal distribution (Z-distribution). Here's how you can approach each part of the question:
(a) To find the probability that x is more than 60, we need to calculate the area under the curve to the right of 60. Since we're given the mean (μ) and standard deviation (σ), we can convert 60 into a Z-score, which represents the number of standard deviations away from the mean.
Z = (x - μ) / σ
Z = (60 - 85) / 25
Z = -1
To find the probability, we can use a Z-table or a calculator. Looking up a Z-table, we find that the probability corresponding to Z = -1 is approximately 0.1587. However, since we want the probability of x being more than 60, we need to subtract this probability from 1:
P(x > 60) = 1 - 0.1587 = 0.8413
(b) Similarly, to find the probability that x is less than 110, we first convert 110 into a Z-score:
Z = (x - μ) / σ
Z = (110 - 85) / 25
Z = 1
Using the Z-table, we find that the probability corresponding to Z = 1 is approximately 0.8413. Therefore:
P(x < 110) = 0.8413
(c) To find the probability that x is between 60 and 110, we need to find the area under the curve between these two values. Using the Z-scores, we can calculate:
Z1 = (60 - 85) / 25 = -1
Z2 = (110 - 85) / 25 = 1
Using the Z-table, we can find the probabilities corresponding to Z1 and Z2:
P(x between 60 and 110) = P(-1 < Z < 1) = P(Z < 1) - P(Z < -1)
= 0.8413 - 0.1587
= 0.6826
(d) To find the probability that x is greater than 140, we need to calculate the area under the curve to the right of 140. Using the Z-score:
Z = (x - μ) / σ
Z = (140 - 85) / 25
Z = 2.2
Using the Z-table, we find that the probability corresponding to Z = 2.2 is approximately 0.9857. Therefore:
P(x > 140) = 1 - 0.9857 = 0.0143
So, the probability of x being greater than 140 is 0.0143.