Question

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

Answer #1

**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,σ^{2})

[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]**

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 ? = 82
and standard deviation ? = 20. Note: After 50
years of age, both the mean and standard deviation tend to
increase. For an adult (under 50) after a...

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 = 83
and standard deviation σ = 21. Note: After 50
years of age, both the mean and standard deviation tend to
increase. For an adult (under 50) after a...

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 = 86
and standard deviation σ = 26. Note: After 50
years of age, both the mean and standard deviation tend to
increase. For an adult (under 50) after a...

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 = 80 and standard deviation σ =
27. Note: After 50 years of age, both the mean and standard
deviation tend to increase. For an adult (under 50) after a...

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 = 89 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...

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 =
26. Note: After 50 years of age, both the mean and standard
deviation tend to increase. For an adult (under 50) after a...

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 = 89 and standard deviation σ =
29. Note: After 50 years of age, both the mean and standard
deviation tend to increase. For an adult (under 50) after a...

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 = 84 and standard deviation σ =
25. Note: After 50 years of age, both the mean and standard
deviation tend to increase. For an adult (under 50) after a...

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 = 80
and standard deviation σ = 21. Note: After 50
years of age, both the mean and standard deviation tend to
increase. For an adult (under 50) after a...

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 = 80 and standard deviation σ =
26. Note: After 50 years of age, both the mean and standard
deviation tend to increase. For an adult (under 50) after a...

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