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 = 56 and estimated standard deviation σ = 25. 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.)

(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?

Answer 1

Solution :

Given that ,

mean = = 56

standard deviation = = 25

a) P(x < 40) = P[(x - ) / < (40 - 56) / 25]

= P(z < -0.64)

Using z table,

= 0.2611

b) n = 2

= = 56

= / n = 25/ 2 = 17.68

The probability distribution of x is approximately normal with μx = 56 and σx = 17.68

P( < 40) = P(( - ) / < (40 - 56) / 17.68)

= P(z < -0.90)

Using z table

= 0.1841

c) n = 3

= = 56

= / n = 25/ 3 = 14.43

The probability distribution of x is approximately normal with μx = 56 and σx = 14.43

P( < 40) = P(( - ) / < (40 - 56) / 14.43)

= P(z < -1.11)

Using z table

= 0.1335

d) n = 5

= = 56

= / n = 25/ 5 = 11.18

The probability distribution of x is approximately normal with μx = 56 and σx = 11.18

P( < 40) = P(( - ) / < (40 - 56) / 11.18)

= P(z < -1.43)

Using z table

= 0.0764

e) yes,