Question

Let *x* be a random variable that represents the level of
glucose in the blood (milligrams per deciliter of blood) after a
12-hour fast. Assume that for people under 50 years old, *x*
has a distribution that is approximately normal, with mean
*μ* = 57 and estimated standard deviation *σ* = 34. A
test result *x* < 40 is an indication of severe excess
insulin, and medication is usually prescribed.

(a) What is the probability that, on a single test, *x*
< 40? (Round your answer to four decimal places.)

What is the probability that x < 40? (Round your answer to four
decimal places.)

(b) Suppose a doctor uses the average x for two tests taken about a
week apart. What can we say about the probability distribution of
x? *Hint*: See Theorem 7.1.

The probability distribution of x is not normal.The probability
distribution of x is approximately normal with μ_{x} = 57
and σ_{x} = 17.00. The probability
distribution of x is approximately normal with μ_{x} = 57
and σ_{x} = 24.04.The probability distribution of x is
approximately normal with μ_{x} = 57 and σ_{x} =
34.

(c) Repeat part (b) for *n* = 3 tests taken a week apart.
(Round your answer to four decimal places.)

(d) Repeat part (b) for *n* = 5 tests taken a week apart.
(Round your answer to four decimal places.)

Answer #1

a)

mean = 57 , sigam = 34

P(x< 40)

= P(z< (x -mean)/sigma)

= P(z< (40 -57)/34)

= P(z< -0.50)

= 0.3085

b)

n = 2

mean = 57 , std.dev = sigma/sqrt(n)

= 34/sqrt(2) = 24.04

The probability distribution of x is approximately normal with μx = 57 and σx = 24.04

P(x< 40)

= P(z< (x -mean)/sigma)

= P(z< (40 -57)/24.04)

= P(z< -0.71)

= 0.2397

c)

n = 3

mean = 57 , std.dev = sigma/sqrt(n)

= 34/sqrt(3) = 19.6299

The probability distribution of x is approximately normal with μx = 57 and σx = 19.63

P(x< 40)

= P(z< (x -mean)/sigma)

= P(z< (40 -57)/19.63)

= P(z< -0.87)

= 0.1932

d)

n= 5

mean = 57 , std.dev = sigma/sqrt(n)

= 34/sqrt(5) = 15.21

The probability distribution of x is approximately normal with μx =
57 and σx = 15.21

P(x< 40)

= P(z< (x -mean)/sigma)

= P(z< (40 -57)/15.21)

= P(z< -1,12)

= 0.1319

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