3. The amount of caffeine in an energy drink is approximately normally distribut

Question

3. The amount of caffeine in an energy drink is approximately normally distributed with a mean of 101 mg, and a standard deviation of 37.4. Determine: a). The proportion of the energy drinks that are expected to contain less than 71 mg of caffeine. b). The proportion of the energy drinks that are expected to contain between 65 and 125mg of caffeine c). The percent of the energy drinks that are expected to contain greater than 150 mg of caffeine. d). The percentile rank of the energy drinks that are expected to contain 75mg e). The minimum amount of caffeine expected to be in the top 1% of the distribution of caffeine in an energy drink.

Explanation / Answer

Result:

a).

z value for 71, z = ( 71-101)/37.4 = -0.80

P( x < 71) = P( z < -0.80)

=0.2119

b).

z value for 65, z = (65-101)/37.4 = -0.96

z value for 125, z = ( 125-101)/37.4 = 0.64

P( 65<x<125) = P( -0.96<z<0.64)

=P( z < 0.64) – P( z < -0.96)

=0.7389 -0.1685

=0.5704

c).

z value for 150, z = ( 150-101)/37.4 = 1.31

P( x >150) = P( z >1.31)

=0.0951

d).

z value for 75, z = ( 75-101)/37.4 = -0.70

P( x< 75) = P( z < -0.70) =0.242

The value 75 is 24Th percentile

e).

z value for top 1% =2.326

x= mean+z*sd = 101+2.326*37.4

=187.992

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