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 = 72 and estimated standard deviation σ = 41. 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.)


(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 6.1.

The probability distribution of x is not normal.The probability distribution of x is approximately normal with μ_x = 72 and σ_x = 20.50.    The probability distribution of x is approximately normal with μ_x = 72 and σ_x = 28.99.The probability distribution of x is approximately normal with μ_x = 72 and σ_x = 41.

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


(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.)


(e) Compare your answers to parts (a), (b), (c), and (d). Did the probabilities decrease as n increased?

YesNo    

Answer 1

a)

Here, μ = 72, σ = 41 and x = 40. We need to compute P(X <= 40). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (40 - 72)/41 = -0.78

Therefore,
P(X <= 40) = P(z <= (40 - 72)/41)
= P(z <= -0.78)
= 0.2177

b)

The probability distribution of x is approximately normal with μx = 72 and σx = 28.99.

Here, μ = 72, σ = 28.99 and x = 40. We need to compute P(X <= 40). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (40 - 72)/28.99 = -1.1

Therefore,
P(X <= 40) = P(z <= (40 - 72)/28.99)
= P(z <= -1.1)
= 0.1357

c)

Here, μ = 72, σ = 23.67 and x = 40. We need to compute P(X <= 40). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (40 - 72)/23.67 = -1.35

Therefore,
P(X <= 40) = P(z <= (40 - 72)/23.67)
= P(z <= -1.35)
= 0.0885

d)

Here, μ = 72, σ = 18.34 and x = 40. We need to compute P(X <= 40). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (40 - 72)/18.34 = -1.74

Therefore,
P(X <= 40) = P(z <= (40 - 72)/18.34)
= P(z <= -1.74)
= 0.0409

e)
yes