Question

A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 83 and standard deviation σ = 27. Note: After 50 years of age, both the mean and standard deviation tend to increase.

1. For an adult (under 50) after a 12-hour fast, find the probability that x is between 60 and 110. (Round your answers to four decimal places.)

Answer 1

The probability that x is between 60 and 110.

P(60 < x<110) = 0.6436

Explanation:

Step( i ) :-

Given data the random variable 'X' will have a distribution that is approximately normal with mean μ = 83 and standard deviation σ = 27.

Mean of the Population 'μ' = 83

Standard deviation of the Population 'σ' = 27.

Step(ii):-

Given X =60

`Z_(1) = (x-mean)/(S.D) = (60-83)/(27) = -0.851`

Step(iii):-

Given X = 110

`Z_(1) = (x-mean)/(S.D) = (110-83)/(27) = 1`

The Probability that between 60 and 110

P(60 < x<110) = P( -0.851 < z< 1)

= P( Z≤1) - P(Z≤ -0.851)

= (0.5 + A(1)) - (0.5- A(-0.851))

= (0.5 +0.3413)- (0.5 - 0.3023) ( check normal table yellow mark)

= 0.8413 - 0.1977

P(60 < x<110) = 0.6436

Final answer:-

The probability that x is between 60 and 110.

P(60 < x<110) = 0.6436