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 58 ounces and a standard deviation of 6 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 52 and 70 ounces? %
c) what percentage of the widget weights lie above 40? %
question help: video message instructor post to forum

Explanation:

Step1: Recall Empirical Rule for 95%

The Empirical Rule states that for a normal - distribution, 95% of the data lies within 2 standard deviations of the mean.

Step2: Calculate lower bound for 95%

The mean $\mu = 58$ ounces and the standard deviation $\sigma=6$ ounces. The lower bound is $\mu - 2\sigma=58 - 2\times6=58 - 12 = 46$ ounces.

Step3: Calculate upper bound for 95%

The upper bound is $\mu + 2\sigma=58+2\times6=58 + 12 = 70$ ounces.

Step4: Analyze range 52 - 70 ounces

$52=\mu-\sigma$ and $70=\mu + 2\sigma$. The percentage of data between $\mu-\sigma$ and $\mu + 2\sigma$ is $\frac{68 + 95}{2}=81.5\%$ (since the percentage between $\mu-\sigma$ and $\mu+\sigma$ is 68% and between $\mu - 2\sigma$ and $\mu+2\sigma$ is 95%).

Step5: Analyze data above 40 ounces

$40=\mu - 3\sigma$. The percentage of data below $\mu - 3\sigma$ is 0.5%. So the percentage of data above 40 ounces is $100 - 0.5=99.5\%$.

Answer:

a) 46, 70
b) 81.5
c) 99.5