Question

the acme company manufactures widgets. the distribution of widget weights is bell - shaped. the widget weights have a mean of 47 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) 95% of the widget weights lie between and b) what percentage of the widget weights lie between 37 and 67 ounces? % c) what percentage of the widget weights lie above 17? %

Explanation:

Step1: Recall the Empirical Rule

For a normal - distribution, about 95% of the data lies within 2 standard deviations of the mean.

Step2: Calculate the bounds for 95% of the data

The mean $\mu = 47$ and the standard deviation $\sigma=10$. The lower bound is $\mu - 2\sigma=47-2\times10 = 27$ and the upper bound is $\mu + 2\sigma=47 + 2\times10=67$.

Step3: Analyze the range 37 - 67

First, note that $37=\mu - \sigma$ and $67=\mu+2\sigma$. The percentage of data between $\mu-\sigma$ and $\mu + \sigma$ is about 68%, and between $\mu - 2\sigma$ and $\mu+2\sigma$ is about 95%. The percentage between $\mu-\sigma$ and $\mu + 2\sigma$ is $\frac{68 + 95}{2}=81.5\%$.

Step4: Analyze the value 17

Since $17=\mu - 3\sigma$, the percentage of data below $\mu - 3\sigma$ is about 0.15% (from the Empirical Rule). So the percentage of data above 17 is $100 - 0.15=99.85\%$.

Answer:

a) 27, 67
b) 81.5
c) 99.85