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 = 75 and estimated standard deviation σ = 31. 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(mean) < 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 = = 75

standard deviation = = 31

a) P(x < 40) = P[(x - ) / < (40 - 75) / 31]

= P(z < - 1.13)

Using z table,

= 0.8708

b) n = 2

= = 75

= / n = 31/ 2 = 21.92

The probability distribution of x is approximately normal with μx = 75 and σx = 21.92

P( < 40) = P(( - ) / < (40 - 75) / 21.92)

= P(z < - 1.59)

Using z table

= 0.0559

c) n = 3

= = 75

= / n = 31/ 3 = 17.89

The probability distribution of x is approximately normal with μx = 75 and σx = 17.89.

P( < 40) = P(( - ) / < (40 - 75) / 17.89)

= P(z < - 1.96)

Using z table

= 0.0250

d) n = 5

= = 75

= / n = 31/ 5 = 13.86

The probability distribution of x is approximately normal with μx = 75 and σx = 13.86

P( < 40) = P(( - ) / < (40 - 75) / 13.86)

= P(z < - 2.53)

Using z table

= 0.0057