3.2 measures of spread: empirical rule\nthe acme company manufactures widgets. the distribution of widget…

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