Question

A person's blood glucose level and diabetes are closely related. Let x be a random variable...

A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 82 and standard deviation σ = 23. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.)

(a) x is more than 60
1

(b) x is less than 110
2

(c) x is between 60 and 110
3

(d) x is greater than 125 (borderline diabetes starts at 125)
4

Before we go on to solve the problems let us know a bit about Normal Distribution and its properties.

Normal Distribution

A continuous random variable X is said to have a normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Notation:

Standard Normal Distribution

A continuous random variable X is said to have a standard normal distribution if its PDF(Probability Density Function) is given by

its CDF(Cumulative Distribution Function) is given by,

Exact evaluation of ?(x) is not possible but numerical method can be applied. The values of ?(x) has been tabulated extensively in Biometrika Volume I.

Notation:

Property

If X~Normal(μ,σ2)

[By transformation]

Now for a continuous random variable X and any real number 'a',

Coming back to our problem,

Given that,

A person's blood glucose level and diabetes are closely related and

X=milligrams of glucose per deciliter (1/10 of a liter) of blood.

It is given that after a 12 hour fast, the random variable X have a distribution that is approximately normal with mean μ= 82 and standard deviation σ = 23.

(a) Here we need to find the probability that X is more than 60.

[Z~Normal(0,1)]

[μ= 82 and σ = 23]

Now by property of ?(x),

[From Biometrika Tables Vol I]

(b) Here we need to find the probability that X is less than 110.

[X is a continuous distribution]

[Z~Normal(0,1)]

[μ= 82 and σ = 23]

[From Biometrika Tables Vol I]

(c) Here we need to find the probability that X is between 60 and 110.

[X is a continuous distribution]

[Z~Normal(0,1)]

[μ= 82 and σ = 23]

[From Biometrika Tables Vol I]

(d) Here we need to find the probability that X is greater than 125.

[Z~Normal(0,1)]

[μ= 82 and σ = 23]

[From Biometrika Tables Vol I]