Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 63 ounces and a standard deviation of 9 ounces. Use the Standard Deviation Rule, also known as the Empirical Rule. Suggestion: sketch the distribution to answer these questions.

a) 68% of the widget weights lie between
b) What percentage of the widget weights lie between 45 and 72 ounces?
c) What percentage of the widget weights lie above 36 ounces?

A. a) 54−72, b) 68%, c) 97.5%
B. a) 54−72 b) 68%, c) 2.5%
C. a) 54−72 b) 95%, c) 2.5%
D. a) 54−72 b) 95%, c) 97.5%

Answer 1

The Acme Company manufactures widgets with a bell-shaped distribution of weights. Using the Standard Deviation Rule, we can determine the percentages of widget weights that fall within certain ranges. For this specific scenario, approximately 68% of the widget weights lie between 54 and 72 ounces, the percentage of widget weights between 45 and 72 ounces is 68%, and the percentage of widget weights above 36 ounces is 68%.

Step-by-step explanation:

To answer this question, we can use the Standard Deviation Rule, also known as the Empirical Rule. This rule states that for a bell-shaped distribution:

1. Approximately 68% of the data falls within one standard deviation of the mean. In this case, one standard deviation is 9 ounces, so 68% of the widget weights lie between 63 - 9 and 63 + 9, which is 54 - 72 ounces. So, option A: a) 54-72 is correct.
2. Approximately 95% of the data falls within two standard deviations of the mean. Using the same logic as above, two standard deviations is 2 * 9 = 18 ounces. So, 95% of the widget weights lie between 63 - 18 and 63 + 18, which is 45 - 81 ounces. However, we need to find the percentage between 45 and 72 ounces specifically. So we subtract the percentage below 45 ounces (which is half of 68%) from the percentage below 72 ounces (which is 68%). 68% - (68%/2) = 68% - 34% = 34%. So, option B: b) 68% is correct.
3. To find the percentage of widget weights above 36 ounces, we need to find the percentage below 36 ounces and subtract it from 100%. Since we know that 68% of the data falls within one standard deviation of the mean (54-72 ounces), we can find the percentage below 36 ounces by subtracting 68% from 100%. 100% - 68% = 32%. So, the percentage of widget weights above 36 ounces is 100% - 32% = 68%. Option C: c) 2.5% is incorrect.