The Acme Company manufactures widgets

Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 51 ounces and a standard deviation of 7 ounces.

Use the Empirical Rule, find

a) 68% of the widget weights lie between  and 

b) What percentage of the widget weights lie between 37 and 58 ounces?  %

c) What percentage of the widget weights lie below 72 ?  %

Suggestion: sketch the distribution in order to answer these questions.

Answer 1

a) Per the Empirical Rule 68% of the data lies within 1 standard deviation of the mean, so the weights would lie between the mean +/- 1 standard deviation or 51 +/- 7 = 44 to 58

b) 51 - 37 = 14, which is 2 standard deviations below the mean and 58 - 51 = 7, which is 1 standard deviation above the mean, so we need to use the Empirical Rule to find the percentage of the area under the normal curve between the mean - 2 standard deviations and the mean + 1 standard deviations which would be 13.5 + 34 + 34 = 81.5%

c) 72 - 51 = 21, which is 3 standard deviations above the mean. Per the Empirical Rule, there is 99.7% of the data between the mean + 3 standard deviations and the mean - 3 standard deviations. There is 0.3% total at the tails beyond 3 standard deviations, so there is 0.15% below the mean - 3 standard deviations. So in total there is 99.7 + 0.15 = 99.85% below 72.