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 = 50. 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 approximately normal with μ_x = 75 and σ_x = 25.00.The probability distribution of x is approximately normal with μ_x = 75 and σ_x = 35.36.     The probability distribution of x is not normal.The probability distribution of x is approximately normal with μ_x = 75 and σ_x = 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 = = 75

standard deviation = = 50

a) P(x < 40 ) = P[(x - ) / < ( 40 - 75 ) / 50 ]

= P(z < -0.7)

Using z table

= 0.2420

b) n = 2

=    = 75

= / n = 50 / 2 = 35.36

The probability distribution of x is approximately normal with = 75 and = 35.36

P( < 40 ) = P(( - ) / < ( 40 - 75 ) / 35.36 )

= P(z < -0.99)

Using z table

= 0.1611

c) n = 3

= 75

= / n = 50/ 3 = 28.87

P( < 40) = P(( - ) / < ( 40 - 75 ) / 28.87)

= P(z < -1.21)

Using z table

= 0.1131

d) n = 5

= 75

= / n = 50/ 5 = 22.36

P( < 40) = P(( - ) / < ( 40 - 75 ) / 22.36 )

= P(z < -1.57)

Using z table

= 0.0582