Question

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 5 ounces.

Use the Standard Deviation Rule, also known as the Empirical Rule. Suggestion: sketch the distribution in order to answer these questions.

a) 68% of the widget weights lie between_____ oz and _____OZ

b) What percentage of the widget weights lie between 44 and 59 ounces? _____%

c) What percentage of the widget weights lie above 39 ?________%​

Answer 1

a) 49 oz and 59 oz

b) 81.8 %

c) 99.9 %

Explanation:

`X \sim N(54, 5^2)`

The Empirical Rule states that:

• 68% of the data falls within one standard deviation from the mean
• 95% of the data falls within two standard deviations from the mean
• 99.7% of the data falls within three standard deviations from the mean

a)

`\mu - \sigma=54-5=49`

`\mu + \sigma=54+5=59`

⇒ 68% of the widget weights lie between 49 oz and 59 oz

b) 59 oz is 1 standard deviation from the mean

44 oz is 2 standard deviations from the mean

Therefore, 84.1 - 2.3 = 81.8 %

c) 39 is 3 standard deviations from the mean

Therefore, 100 - 0.1 = 99.9 %

**Please find attached a sketch of the distribution. The mean is shown with the solid line and the standard deviations with dash lines**