Question

A patient is classified as having gestational diabetes if their average glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Rebecca's doctor is concerned that she may suffer from gestational diabetes. There is variation both in the actual glucose level and in the blood test that measures the level. Rebecca's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ= 146 mg/dl and σ = 7 mg/dl. Using the Central Limit Theorem, determine the probability of Rebecca being diagnosed with gestational diabetes if her glucose level is measured: a) Once? b) 8 times c) 10 times

Answer 1

Solution :

Given that ,

mean = $\mu$ = 146

standard deviation = $\sigma$ = 7

a) P(x > 140) = 1 - p( x< 140)

=1- p P[(x - $\mu$ ) / $\sigma$ < (140 - 146) / 7]

=1- P(z < -0.86)

Using z table,

= 1 - 0.1949

= 0.8051

b) n = 8

$\mu$_{$\bar x$} = $\mu$ = 146

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 7 / $\sqrt$ 8 = 2.475

P($\bar x$ > 140) = 1 - P($\bar x$ < 140)

= 1 - P[($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (140 - 146) / 2.475]

= 1 - P(z < -2.42)

Using z table,    

= 1 - 0.0078

= 0.9922

c) n = 10

$\mu$_{$\bar x$} = $\mu$ = 146

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 7 / $\sqrt$ 10 = 2.214

P($\bar x$ > 140) = 1 - P($\bar x$ < 140)

= 1 - P[($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (140 - 146) / 2.214]

= 1 - P(z < -2.71)

Using z table,    

= 1 - 0.0034

= 0.9966