The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 57 ounces and a standard deviation of 6 ounces. Use the Empirical Rule and a sketch of the normal distribution in order to answer these questions. a) \( 95 \% \) of the widget weights lie between b) What percentage of the widget weights lie between 39 and 69 ounces? c) What percentage of the widget weights lie above 51 ?

To answer your questions using the Empirical Rule, let’s first recap the key points: In a normal distribution, about 68% of the data lies within one standard deviation from the mean, about 95% within two standard deviations, and about 99.7% within three standard deviations. a) To find the range for \( 95 \% \) of the widget weights, we calculate two standard deviations from the mean: - Lower limit: \( 57 - 2(6) = 45 \) ounces - Upper limit: \( 57 + 2(6) = 69 \) ounces Thus, \( 95 \% \) of the widget weights lie between 45 ounces and 69 ounces. b) The range from 39 to 69 ounces encompasses less than \( 95 \% \). - The lower limit (39 ounces) is three standard deviations below the mean: \( 57 - 3(6) = 39 \). - Since \( 99.7\% \) of weights are within this limit, and we've already established that \( 95 \% \) is between 45 and 69, we conclude that \( 49.7\% \) of the weights lie between 39 and 69 ounces. c) To find the percentage above 51 ounces, first calculate how many standard deviations 51 is from the mean: - \( 51 - 57 = -6 \) ounces, which means it's one standard deviation below the mean. Using the Empirical Rule, since \( 84 \% \) of the data lies below one standard deviation (51 ounces), that means \( 16 \% \) of the widget weights lie above \( 51 \) ounces.