Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 50 ounces and a standard deviation of 5 ounces. Use the Standard Deviation Rule, also known as the Empirical Rule. Suggestion: sketch the distribution in order to answer these questions. a) 95% of the widget weights lie between and b) What percentage of the widget weights lie between 45 and 60 ounces? % c) What percentage of the widget weights lie below 65 ? %

Answer 1

The Empirical Rule applies to a normal, bell-shaped curve and states that

• within one standard deviation of the mean (both left-side and right-side) there is about 68% of the data;
• within two standard deviations of the mean (both left-side and right-side) there is about 95% of the data; and
• within three standard deviations of the mean (both left-side and right-side) there is about 99.7% of the data.

a)

;

By empirical rule above, 95% of the widget weights lie between 40 and 60.

Note that there can can be more than such intervals where, P(A<X<B)=0.95.

Another way,

b)

# 81.86 % of the widget weights lie between 45 and 60 ounces.

c)

#99.86 % of the widget weights lie below 65.

R-commands:

a)
> round(pnorm(60,mean=50,sd=5)-pnorm(40,mean=50,sd=5),2)
[1] 0.95

> qnorm(0.025)
[1] -1.959964
> qnorm(0.975)
[1] 1.959964
> qnorm(0.025,mean=50,sd=5)
[1] 40.20018
> qnorm(0.975,mean=50,sd=5)
[1] 59.79982
> round(qnorm(0.025,mean=50,sd=5),2)
[1] 40.2
> round(qnorm(0.975,mean=50,sd=5),2)
[1] 59.8

> 1.96*5
[1] 9.8
> 9.8-50
[1] -40.2
> pnorm(59.8,mean=50,sd=5)
[1] 0.9750021
> pnorm(40.2,mean=50,sd=5)
[1] 0.0249979

pnorm(59.8,mean=50,sd=5)-pnorm(40.2,mean=50,sd=5) #Cross-check
[1] 0.9500042
#95% of the widget weights lie between 40.2 and 59.8.

b)
> pnorm(45,mean=50,sd=5)
[1] 0.1586553
> pnorm(60,mean=50,sd=5)
[1] 0.9772499
> pnorm(60,mean=50,sd=5)-pnorm(45,mean=50,sd=5)
[1] 0.8185946

> pnorm(2)
[1] 0.9772499
> pnorm(-1)
[1] 0.1586553
> pnorm(2)-pnorm(-1)
[1] 0.8185946
# 81.86 % of the widget weights lie between 45 and 60 ounces.

c)
> pnorm(65,mean=50,sd=5)
[1] 0.9986501
> pnorm(3)
[1] 0.9986501
#99.86 % of the widget weights lie below 65.