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