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Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped with a mean of 59 ounces and a standard deviation of 9 ounces. Using the Empirical Rule, answer the following questions. Suggestion: Sketch the distribution.

a) 68% of the widget weights lie between  and 

b) What percentage of the widget weights lie between 32 and 68 ounces?  %

c) What percentage of the widget weights lie below 77?  %

Answer 1

# of standard deviations above/below the mean x-score

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-3 59-9-9-9=32 <--- lower fence

-2 59-9-9 = 41

-1 59-9 = 50

0 59

+1 59+9 = 68

+2 59+9+9 = 77

+3 59+9+9+9 = 86 <-- upper fence

Empirical rules says:

68% or roughly 2/3 of the data lies within 1 standard deviation from the mean which is between 50 and 68

95% of the data lies within 2 standard deviations from the mean which is between 41 and 77

98% of the data lies within 3 standard deviations form the mean which is between 32 and 86

50% or half of the data lies below the mean, which is 0 - 59, and the upper half is 59 - infinity

a) As stated above, 68% or roughly 2/3 of the data, lies within 1 standard deviation of the mean

which is 50 and 68

b) roughly 2/3 of the data lies within 1 standard deviation of the mean, which is 50 to 68...

so 1/3 of the data lies between 59 to 68 by symmetric property of bell curve...

1/2 of the data lies below 59, which includes the lower fence

1% of the data lies below the lower fence

1/2 + 1/3 - 1% = 1/2 + 1/3 - 1/100 = 300/600 + 200/600 - 6/600 = 494/600 = 82 + 1/3 % = 82.3333... %

c) 95% of the data lies within 2 standard deviations of the mean, which is between 41 and 77;

so 47.5% of the data lies between 59 and 77, by symmetric property of the bell curve

50% of the data lies below 59...

so 50% + 47.5% = 97.5%