The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 56 ounces and a standard deviation of 9 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 29 b) What percentage of the widget weights lie between 47 and 83 ounces? 83.9 c) What percentage of the widget weights lie below 74 ?

The Empirical Rule is like a magic wand for bell-shaped distributions! It tells us that almost all (99.7%) of the data falls within three standard deviations from the mean. Since the mean weight is 56 ounces and the standard deviation is 9 ounces, we can calculate that weights will fall within the range of 29 ounces (56 - 3*9) to 83 ounces (56 + 3*9). For your second query, the interval of 47 to 83 ounces captures a little more than two standard deviations above the mean, which according to the Empirical Rule, includes about 95% of the data, as it encompasses from two standard deviations below (38 ounces) to the mean and then up to three standard deviations above (83 ounces). To find the percentage of widget weights that lie below 74 ounces, first find how many standard deviations 74 is from the mean: \( (74 - 56) / 9 \approx 2 \). This means 74 is two standard deviations above the mean. According to the Empirical Rule, about 95% of the weights fall below this point. So, it's safe to say that roughly 97.5% of the widget weights lie below 74 ounces!