Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 58 ounces and a standard deviation of 11 ounces.

Use the Standard Deviation Rule, also known as the Empirical Rule.

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

a) 68% of the widget weights lie between  and

b) What percentage of the widget weights lie between 25 and 69 ounces?  %

c) What percentage of the widget weights lie above 36 ?  %

Answer 1

Let the random variable X is denoted as widget weight.

Given : The distribution of X is bell shaped with mean 58 and standard deviation 11.

i.e.

Empirical Rule or three sigma rule.

68.27% value lies within one standard deviation.

95.45% values lies within two standard deviation.

99.73 % value lies within three standard deviation.

i.e.

i.e.

Graph:

a) By empirical rule

68% of the widget weight lie between 47 and 69 ounces

b ) Required Percentage = P ( 25 < X < 69) = P ( 25 < X < 58) + P (58 <X < 69)

From graph

P(25 < X < 58 ) = 34.1% + 34.1%

P( 58 < X < 69) = 13.6%

Hence required probability is

P ( 25 < X < 69) = 34.1% + 34.1% + 13.6% = 81.8%

81.8% percentage of widget weights lies between 25 and 69 ounces.

c) Required Percentage = P ( X > 36)

= 1- P (X < 36)

By empirical rule 95.45% widget weight lies between 36 and 80.

Since the distribution is symmetric.

2.275% ( 1 -0.9545 = 0.455, 0.455/2 = 0.0275 ) widget weight below 36 and above 80.

Hence required percentage

P ( X > 36) = 1- 0.02275 = 0.97725 = 97.725%

97.725 % widget weight lie above 36 ounces.