SOLUTION: a manufacturing process produces items whose weights are normally distributed . it is known that 22.57% of all the items produced weigh between 100 grams up to the mean and 49.18 %

Question 1113488: a manufacturing process produces items whose weights are normally distributed . it is known that 22.57% of all the items produced weigh between 100 grams up to the mean and 49.18 % weigh from the mean up to 190 grams . Determine the mean and the standard deviation
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
Probability (P) (100 < X < mean) = P( X < mean) - P( X < 100 )
:
0.2257 = P( X < mean) - P( X < 100 )
:
P( X < 100 ) = 0.50 - 0.2257 = 0.2743
:
z-score for 0.2743 = -0.60
:
1) (100 - mean) / standard deviation = -0.60
:
P ( mean < X < 190 ) = P ( X < 190 ) - P ( X < mean )
:
0.4918 = P ( X < 190 ) - 0.50
:
P ( X < 190 ) = 0.9918
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z-score associated with 0.9918 = 2.4
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2) (190 - mean) / standard deviation = 2.4
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solve equation 1 for mean
:
mean = 100 + (standard deviation * 0.60)
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substitute for the mean in equation 2
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190 -100 -standard deviation * 0.60 = 2.4 * standard deviation
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3 * standard deviation = 90
:
standard deviation = 30
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we can use either equation 1 or equation 2 to find the mean, pick equation 2
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(190 - mean) / 30 = 2.4
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190 - mean = 72
:
mean = 190 -72 = 118
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mean is 118 and standard deviation = 30
:
check with equation 1
:
(100 -118)/30 = -0.60
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-18/30 = -0.60
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-0.60 = -0.60
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answer checks
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