Determining whether a five minute wait is unusual

Expert intro: This question asks you to judge whether 5 minutes is unusually long by comparing it to the distribution or benchmark given in the problem.

Detailed walkthrough

Key idea

To decide whether five minutes is unusually long, you must compare it with the expected waiting time from the original context. In statistics and probability, a value is usually called unusual if it is far from the center of the distribution, often using a rule such as being more than 2 standard deviations above the mean, or having a very small probability of occurring.

If the problem gave you data about waiting times, the first step is to identify the appropriate reference point: the mean, median, or a probability model. Without that context, “five minutes” cannot automatically be labeled unusual.

How to judge unusual values

If the waiting time data are approximately normal, a common rule is:

• compute the z-score:
  $z = \frac{x-\mu}{\sigma}$
• interpret the result:
  • if $|z| > 2$, the value is often considered unusual
  • if the probability of waiting at least that long is very small, it is also unusual

For example, if the average wait is 2 minutes with a standard deviation of 1 minute, then 5 minutes gives

$z = \frac{5-2}{1} = 3,$

which is far above the mean and would be unusual.

What the wording is really asking

The phrase “Is five minutes an unusually long time to wait?” is not asking for a calculation by itself. It is asking for a comparison against the distribution from Part a or from the given data table. In a complete solution, you should state the criterion you used and then conclude whether 5 minutes falls beyond that threshold.

If the waiting time distribution shows that 5 minutes is in the tail, then yes, it is unusual. If 5 minutes is near the center of the distribution, then no, it is not unusual.

Common interpretation rule

A strong answer should mention one of these accepted ideas:

• z-score rule: unusually high if more than 2 standard deviations above the mean
• tail probability rule: unusual if the probability is less than 0.05 or similarly small
• context rule: unusual only relative to the actual data or model provided

That is why the correct response depends on the original statistics information from the question, not on the number 5 alone.

💡 Pitfall guide

A common mistake is to answer “yes” just because 5 sounds large. In statistics, a value is unusual only relative to the data set or model. Another mistake is using the median or mean alone without checking spread. A wait time of 5 minutes may be ordinary if the distribution is wide, but unusual if most waits cluster around 1 or 2 minutes. Always reference the rule or calculation given in the problem.

🔄 Real-world variant

If the question changed to: “Is five minutes an unusually long time to wait if the mean is 3 minutes and the standard deviation is 1 minute?” then you would compute a z-score: $z=(5-3)/1=2.$ That is right on the border of being unusual. If the question changed again to use a skewed waiting-time distribution, you would rely more on the percentile or tail probability than on the z-score rule. The conclusion can change when the model changes.

🔍 Related terms

z-score, standard deviation, tail probability