Question

3.2 measures of spread: empirical rule
the acme company manufactures widgets. the distribution of widget weights is bell - shaped. the widget weights have a mean of 54 ounces and a standard deviation of 10 ounces.
use the standard deviation rule, also known as 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 34 and 84 ounces? %

Explanation:

Step1: Recall the Empirical Rule

The Empirical Rule states that for a bell - shaped distribution: about 68% of the data lies within 1 standard deviation of the mean, about 95% lies within 2 standard deviations of the mean, and about 99.7% lies within 3 standard deviations of the mean.

Step2: Calculate the bounds for 99.7% of the data

The mean $\mu = 54$ ounces and the standard deviation $\sigma=10$ ounces. The lower bound is $\mu - 3\sigma=54 - 3\times10=24$ ounces. The upper bound is $\mu + 3\sigma=54+3\times10 = 84$ ounces.

Step3: Calculate the number of standard deviations for part b

For $x_1 = 34$ ounces, the number of standard deviations from the mean is $z_1=\frac{34 - 54}{10}=\frac{- 20}{10}=-2$. For $x_2 = 84$ ounces, the number of standard deviations from the mean is $z_2=\frac{84 - 54}{10}=\frac{30}{10}=3$. The percentage of data between $z=-2$ and $z = 3$ is the percentage of data between 2 standard deviations below the mean and 3 standard deviations above the mean. The percentage of data between $\mu - 2\sigma$ and $\mu+2\sigma$ is 95% and between $\mu - 3\sigma$ and $\mu+3\sigma$ is 99.7%. The percentage of data between $\mu - 2\sigma$ and $\mu+3\sigma$ is $\frac{95+(99.7 - 95)}{2}=97.35\%$.

Answer:

a) 24 ounces, 84 ounces
b) 97.35%