The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 47 ounces and a standard deviation of 8 ounces.

Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 47 ounces and a standard deviation of 8 ounces.

Use the Empirical Rule to answer the questions.

a) 95% of the widget weights lie between ___ and ___  ounces.

b) What percentage of the widget weights lie between 39 and 63 ounces?   %

c) What percentage of the widget weights lie above 23 ounces?   %

Answer 1

Hi Lynsey, the empirical rule is also known as the 68-95-99.7 rule, referring that 68% of values in a normal distribution fall within one standard deviation of the mean, 95% fall within two, and 99.7% fall within +/-3 standard deviations

with mean 47 and standard deviation 8, 95% of values lie between 47-2(8) and 47+2(8) = 31 and 63

b) 47-39 = 8 (one standard deviation); 63-47 = 16 (two standard deviations). We know 95% lies within two standard deviations, and 68% lies within one, so it can be deduced that 95-68=27% of values lie between one and two deviations. Since we have one and a half deviations, we can say that 68% + (27%/2) = 81.5% of values lie between 39 and 63

c) 47-23 = 24/8 = 3. 23oz is the 3rd standard deviation below the mean. If 99.7% of values lie within three standard deviations, then 0.3% lies outside of that range. If we add 99.7% and the (0.3%/2) that lies above the third standard deviation we get .9985%