Question

Suppose USAir accounts for 20% of all U.S domestic flights. As of mid-1194, USAir was involved in four of the previous seven major disasters. "That’s enough to begin getting suspicious but not enough to hang them," said Dr. Brad Efton in The New York Times (September 11, 1994, Sec 4, p. 4). Using a binomial distribution, comment on Dr. Efton’s remark.

Answer 1

To comment on Dr. Efton’s remark using a binomial distribution, one would calculate the probability of USAir being involved in four or more disasters by chance, given they make up 20% of flights. This calculation provides a way to assess if the occurrences are suspiciously high, but other factors would need to be considered for a comprehensive analysis.

Step-by-step explanation:

The question presented is asking for a statistical analysis on whether the occurrence of USAir being involved in four of the last seven major airline disasters as of mid-1994 is suspicious, given that USAir accounts for 20% of all U.S. domestic flights. This scenario is ideal for application of the binomial distribution, which is used to model the number of successful outcomes in a given number of trials when each trial has the same probability of success.

To analyze Dr. Efton's remark using a binomial distribution, one would set up a binomial probability model with the number of trials (n) being seven (the total number of major disasters), the number of successes (x) being four (the number of times USAir was involved), and the probability of success on a single trial (p) being 0.20 (the proportion of U.S domestic flights accounted for by USAir). The question then is whether the observed data (USAir being involved in four out of seven disasters) is highly unlikely if USAir flights were not predisposed to accidents.

Examining this with the binomial distribution would involve calculating the probability of USAir being involved in four or more disasters purely by chance. If this probability is very low, it could suggest that their involvement is not merely random but possibly due to other factors. However, a more in-depth analysis considering other variables and context, beyond what a simple binomial probability calculation can offer, would likely be necessary to draw meaningful conclusions about USAir's safety record.

Answer 2

Using binomial distribution to calculate the probability of USAir's involvement in major air disasters shows whether their incident rate is due to random chance or suggests other factors.

Step-by-step explanation:

Using a binomial distribution to evaluate USAir's involvement in four out of the previous seven major air disasters, given their 20% market share of U.S domestic flights, we can calculate the probability of such an event occurring due to random chance. If we let 'success' represent the unfortunate event of a disaster involving USAir, with a probability of success being 0.20 (due to their market share), the binomial probability formula P(X=k) = (n choose k) * p^k * (1-p)^(n-k) can be used, where 'n' is the number of trials (seven disasters), 'k' is the number of successes (four disasters involving USAir), and 'p' is the probability of success on each trial.

Calculating this, we would find the probability of USAir being involved in exactly four disasters purely by chance. If this probability is sufficiently low, it could suggest that there might be other factors at play beyond random chance, supporting Dr. Efton's remark about suspicion. However, if the probability is not exceedingly low, it does not provide strong enough evidence to 'hang them', as Dr. Efton noted, indicating that the number of incidents could still reasonably occur by chance given their share of the market.