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 = 55 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 7.1.

The probability distribution of x is not normal.The probability distribution of x is approximately normal with μ_x = 55 and σ_x = 41.    The probability distribution of x is approximately normal with μ_x = 55 and σ_x = 28.99.The probability distribution of x is approximately normal with μ_x = 55 and σ_x = 20.50.

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.)

Answer 1

Solution :

Given that ,

mean = = 55

standard deviation = = 41

a) P(x < 40) = P[(x - ) / < (40 - 55) / 41]

= P(z < -0.37)

Using z table,

= 0.3557

b) n = 2

= = 55

= / n = 41/ 2 = 28.99

The probability distribution of x is approximately normal with μx = 55 and σx = 28.99.

P( < 40) = P(( - ) / < (40 - 55) / 28.99)

= P(z < -0.52)

Using z table

= 0.3015

c) n = 3

= = 55

= / n = 41/ 3 = 23.67

The probability distribution of x is approximately normal with μx = 55 and σx = 23.67

P( < 40) = P(( - ) / < (40 - 55) / 23.67)

= P(z < -0.63)

Using z table

= 0.2643

d) n = 5

= = 55

= / n = 41/ 5 = 18.34

The probability distribution of x is approximately normal with μx = 51 and σx = 18.34

P( < 40) = P(( - ) / < (40 - 55) / 18.34)

= P(z < -0.82)

Using z table

= 0.2061