When we take a flight from one location to another, we are often frustrated with the restrictions placed on cargo and carry-on luggage. Sometimes we bring a piece of luggage that is 0.5 lbs over the limit, only to be told we need to reduce the weight! Even more frustrating? When we find the flight isn't even full! Sometimes, we bring carry-on luggage onto the plane, only to be told the luggage needs to be checked into cargo due to lack of space in the overhead bins. Occasionally we board the plane and notice there to be plenty of space! While much of this process is attributed to simple economics (every pound reduces fuel economy, after all), an important consideration is the distribution of weight throughout the plane. If we assume that passengers book flights such that their body weight and luggage weight are randomly/normally distributed, then it seems fine that there would be no general inclination for one side of the plane to be significantly heavier than any other side of the plane. However, in our opening case study, this assumption failed - the tail-end of the plane was considerably heavier than the front-end of the plane. This leads us to a conversation of how to control such extremes, such that it is not possible to have such wildly unbalanced distributions of weights on planes. Consider virtually any airline flying domestically in the United States. Most carriers have a weight limit of 50 lbs. per bag that is to be checked into cargo. Additionally, we will assume from available data that the weight of people in the United States is, on average, 172.2 lbs. with a standard deviation of 29.8 lbs. 1. Assuming a group of 100 passengers travel, discuss for this data the worst possible case of weight distribution (leading to a tail-heavy plane). What would the total weight difference between the two ends of the plane be? It is relatively safe to say that almost 100% of people would be within 2 standard deviations of the mean. We now consider how the baggage is distributed throughout the cargo hold. In an ideal setting, a bag weight of 50 lbs, the maximum weight allowed, would be an "extreme" for any given flight. In other words, we would expect most people to have bag weights well below the maximum. Since, however, we are dealing with a normal distribution, we will, at the very least, assume that very few people have bags that have weights near the described maximum. That is, a bag weight of 50 lbs would likely be three standard deviations away from the mean. 2. Suppose the bag weights are found to be normally distributed with a mean weight of 25 lbs. What would the standard deviation of bag weights be? Describe, in words, what this value means. Is it realistic, given what we know about bag weights? Would the standard deviation be more or less realistic than if the mean bag weight was 20 lbs? 34 lbs? 3. In reality, what would you expect the shape of the distribution of bag weights to look like? Would it be normal? Would it be skewed in one direction or another? Would it be bimodal (having two peaks instead of just one)? Use your intuition to propose a reasonable possibility.