Question

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.

Answer 1

Given the weight of people in the United States is, on average, 172.2 lbs. with a standard deviation of 29.8 lbs.

1)Given  100 passengers travel, the sum of weights has mean and the standard deviation is

The total weight difference between the two ends of the plane be, assuming 100% of people would be within 2 standard deviations of the mean is

2) Given the bag weights are found to be normally distributed with a mean weight of 25 lbs. Also a bag weight of 50 lbs would likely be three standard deviations away from the mean. The standard deviation of bag weights is found as

If the mean is 20 lbs, using the above rule of standard devaition,

.

The standard deviation has increased.

If the mean is 34 lbs, using the above rule of standard devaition,

.

The standard deviation has decreased.

3) Since the bag weights (individual) are found to be normally distributed with a mean weight of 25 lbs, the distribution of total bag weights is also normal.