The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 37 ounces and a standard deviation of 3 ounces. Use the Empirical Rule. Suggestion: sketch the distribution in order to answer these questions. a) \( 68 \% \) of the widget weights lie between b) What percentage of the widget weights lie between 31 and 40 ounces? c) What percentage of the widget weights lie above 28 ?

The Empirical Rule indicates that for a bell-shaped distribution, about 68% of the data falls within one standard deviation of the mean. Here, with a mean of 37 ounces and a standard deviation of 3 ounces, 68% of the widget weights lie between 34 ounces (37 - 3) and 40 ounces (37 + 3). For the second question, to find the percentage of widget weights between 31 and 40 ounces, we can apply the Empirical Rule again. We know that 68% is within one standard deviation (34 to 40 ounces). Adding the range from 31 ounces (two standard deviations below the mean) to 34 ounces, we include another 13.5%. Thus, the total percentage between 31 and 40 ounces is approximately 81.5%. To find the percentage of widget weights above 28 ounces, first note that 28 ounces is three standard deviations below the mean (37 - 9). According to the Empirical Rule, about 99.7% of the weights lie within three standard deviations, meaning roughly 0.15% would be below 28 ounces. Therefore, about 99.85% of the widget weights are above 28 ounces.