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 45 b) What percentage of the widget weights lie between 39 and 69 ounces? c) What percentage of the widget weights lie above 51 ? Question Help:

The Empirical Rule states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. For the Acme Company’s widgets, with a mean of 57 ounces and a standard deviation of 6 ounces, this means that 95% of widget weights lie between 45 ounces (57 - 2*6) and 69 ounces (57 + 2*6), perfectly aligning with part a. For part b, since 39 ounces (57 - 3*6) to 69 ounces (57 + 2*6) spans three standard deviations below the mean to two above, using the Empirical Rule, we can find that approximately 99.7% of the widget weights lie between 39 and 69 ounces. Finally, for part c, weights above 51 ounces (the mean) include roughly half of the distribution since the normal distribution is symmetrical. Therefore, approximately 50% of widget weights are above 51 ounces.