Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 55 ounces and a standard deviation of 8 ounces. Use the Empirical Rule. Suggestion: sketch the distribution in order to answer these questions. a) 99.7% of the widget weights lie between and b) What percentage of the widget weights lie between 39 and 79 ounces? % c) What percentage of the widget weights lie below 63 ? %

Answer 1

Answer:

a) 31 and 79 b) 97.35% c) 84%

Step-by-step explanation:

If we are going to use the empirical rule, we should use the normal probability density function. We have a mean of $\mu$=55 ounces and a standard deviation of $\sigma$=8 ounces. Let's compute the following figures

$\mu-\sigma$ = 55-8=47

$\mu-2\sigma$ = 55-2(8)=39

$\mu-3\sigma$ = 55-3(8)=31

$\mu+\sigma$ = 55+8=63

$\mu+2\sigma$ = 55+2(8)=71

$\mu+3\sigma$ = 55+3(8)=79

The empirical rule is shown in the image.

a) 99.7% of the widget weights lie between 31 and 79. 31 is the mean minus 3 standard deviations and 79 is the mean plus 3 standard deviations.

b) Between 39 and 71 lie 95%  of the widget weights, and between 71 and 79 lie 2.35% of the widget weights, therefore, between 39 and 79 lie 95%+2.35%=97.35% of the widget weights.

c) Below 55 lie 50% of the widget weights and between 55 and 63 lie 34% of the widget weights, therefore, below 63 lie 84% of the widget weights.