Finding an unusual sampling probability with the normal curve

Key concept: This problem focuses on the normal curve, right-tail probability, and the 5% unusualness rule. It also tests whether you can connect a computed probability to a statistical conclusion about the assumed mean.

Step by step

Compute the right-tail probability

The problem gives

$P(\bar{x}\ge 6.5) \approx 1 - 0.9560 = 0.0440$.

That value is the area to the right of 6.5 under the sampling distribution.

Decide whether the result is unusual

A result is unusual if its probability is less than 0.05.

Since

$0.0440 < 0.05$,

the result is unusual.

Choose the correct interpretation

Because the probability is below 5%, the result supports the idea that the assumed mean may not be correct. The prompt states that if a result is unusual, it is reasonable to believe the assumed mean is not correct.

So the conclusion is that it is reasonable to believe the mean is greater than 6.

Why the cutoff matters

The 5% rule is a standard decision threshold in introductory statistics. It does not mean the event is impossible; it means the event is rare enough that we start questioning the model.

Here, a sample mean of 6.5 is in the extreme right tail of the distribution, so it is not a typical outcome if the true mean were really 6.

Final interpretation

The correct statistical reading is:

• the probability is 0.0440,
• the event is unusual,
• and the sample provides evidence that the assumed mean is too low.

Pitfall alert

A common mistake is to think that any probability below 0.5 is unusual. That is far too loose. In this topic, unusual means below 0.05, not below 0.50. Another mistake is to stop after computing 0.0440 and forget to interpret it in context. The number itself is not the final answer; the final answer is the decision it supports. Also, students sometimes confuse “mean greater than 6” with “sample mean greater than 6.5.” The problem is about evidence regarding the population or assumed mean, not just restating the sample result.

Try different conditions

If the right-tail probability had been $0.0825$ instead of $0.0440$, the result would not be unusual because it would be above 0.05. A variant question might ask: “Suppose $P(\bar{x}\ge 6.5)=0.0825$. Is the result unusual, and what should you conclude?” In that case, the answer would be that the result is not unusual and there is not enough evidence to say the assumed mean is wrong. The same interpretation rule applies, but the decision flips because the probability crosses the 5% cutoff.

Further reading

right-tail probability, sampling distribution, 5% significance rule