The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 60 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) 99.7% of the widget weights lie between and (b) What percentage of the widget weights lie between 50 and 75 ounces? (c) What percentage of the widget weights lie above 55?

a) 99.7% of the widget weights lie between 45 and 75 ounces

b) 97.2% of the widget weights lie between 50 and 75 ounces

c) 84% of the widget weights lie above 55

Step-by-step explanation:

The Empirical Rule states that:

50% of the values of a measure is above the mean, and the other 50% is below the mean.

99.7% of the values of a measure lie between 3 standard deviations of the mean.

95% of the values of a measure lie between 2 standard deviations of the mean.

68% of the values of a measure lie between 1 standard deviations of the mean.

In this problem, we have that: The widget weights have a mean of 60 ounces and a standard deviation of 5 ounces.

(a) 99.7% of the widget weights lie between

3 standard deviations of the mean, so:

60 - 3*5 = 60 + 3*5 = 45 and 75 ounces

(b) What percentage of the widget weights lie between 50 and 75 ounces?

We have to find the percentage that are below 75 and subtract by the percentage that are below 50. So

75 is 3 standard deviations above the mean. So 99.7% of the measures are below 75.

50 is 2 standard deviations below the mean. So only 5% of the measures that are below the mean are below 50.

So

99.7% - (50%) 5% = 99.7% - 2.5% = 97.2%

(c) What percentage of the widget weights lie above 55?

55 is one standard deviation below the mean.

50% of the widget weights are above the mean.

Of the 50% that is below, 68% lie between one standard deviation (So from 55 to 60)

So

50% + 68% (50%) = 84%