Question

The weight of topsoil sold in a week is normally distributed with a mean of 1600 tons and a standard deviation of 160 tons.

(a) What percentage of weeks will sales exceed 1760 tons? (Round your answer to two decimal places.)
%

(b) What percentage of weeks will sales be less than 1520 tons? (Round your answer to two decimal places.)
%

(c) What percentage of weeks will sales be between 1360 and 1760 tons? (Round your answer to two decimal places.)
%

Part a)
X ~ N ( µ = 1600 , σ = 160 )
P ( X > 1760 ) = 1 - P ( X < 1760 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 1760 - 1600 ) / 160
Z = 1
P ( ( X - µ ) / σ ) > ( 1760 - 1600 ) / 160 )
P ( Z > 1 )
P ( X > 1760 ) = 1 - P ( Z < 1 )
P ( X > 1760 ) = 1 - 0.8413
P ( X > 1760 ) = 0.1587

Percentage = 0.1587 * 100 = 15.87%

Part b)
X ~ N ( µ = 1600 , σ = 160 )
P ( X < 1520 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 1520 - 1600 ) / 160
Z = -0.5
P ( ( X - µ ) / σ ) < ( 1520 - 1600 ) / 160 )
P ( X < 1520 ) = P ( Z < -0.5 )
P ( X < 1520 ) = 0.3085

Percentage = 0.3085 * 100 = 30.85%

Part c)
X ~ N ( µ = 1600 , σ = 160 )
P ( 1360 < X < 1760 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 1360 - 1600 ) / 160
Z = -1.5
Z = ( 1760 - 1600 ) / 160
Z = 1
P ( -1.5 < Z < 1 )
P ( 1360 < X < 1760 ) = P ( Z < 1 ) - P ( Z < -1.5 )
P ( 1360 < X < 1760 ) = 0.8413 - 0.0668
P ( 1360 < X < 1760 ) = 0.7745

Percentage = 0.7745 * 100 = 77.45%

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