Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 57 ounces and a standard deviation of 8 ounces. Use the Standard Deviation Rule, also known as the Empirical Rule. A) 95% of the widget weights lie between_____and____ B)What percentage of the widget weights lie between 33 and 73 ounces C)What percentage of the widget weights lie above 49

Answer 1

Answer with explanation:

A) The empirical rule says that the 95% of the data falls in between two standard deviations of the mean.

Given : The distribution of widget weights is bell-shaped. i.e. the distribution is a normal distribution.

Population  mean : $\mu=57\text{ ounces}$

Standard deviation : $\sigma= 8\text{ ounces}$

Then, by empirical rule , 95% of data falls in between $\mu\pm 2\sigma$

i.e. $57\pm2(8)=57\pm16=(57-16,57+16)=(41,\ 73)$

Hence, the 95% of the widget weights lie between 41 ounces and 73 ounces .

B) Let x be the random variable that represents the weight of the widgets.

Formula of Z-score : $z=\dfrac{x-\mu}{\sigma}$   (1)

Put x= 33 in (1)

$z=\dfrac{33-57}{8}=-3$

Put x= 73 in (1)

$z=\dfrac{73-57}{8}=2$

Using the standard normal distribution table , we have

P-value : $P(-3<z<2)=P(z<2)-P(z<-3)=0.9772-0.0013=0.9759=97.59\%$

Hence, 97.59% of the widget weights lie between 33 and 73 ounces .

C) Put x= 49 in (1)

$z=\dfrac{49-57}{8}=-1$

P-value : $P(z<-1)=1-P(z\leq-1)=1-0.1587=0.8413=84.13\%$

Hence, 84.13% of the widget weights lie above 49.