Interpreting an unusual sample mean in hypothesis testing

Key concept: This problem uses sampling distribution logic to judge whether a sample mean is unusual. The key ideas are probability interpretation, the 5% rule, and connecting that probability to a reasonable statistical conclusion.

Step by step

Understand the probability statement

We are told that

$P(\bar{x}\ge 2.2)=0.1867$.

This means there is an 18.67% chance of getting a sample mean of at least 2.2 insect fragments per 10-gram portion if the assumed population mean is correct.

Decide whether the result is unusual

A result is considered unusual only if its probability is less than 5%.

Since

$0.1867 > 0.05$,

the result is not unusual.

So the correct choice is:

B. This result is not unusual because its probability is large.

Interpret the conclusion

When a sample outcome is not unusual, it is not strong evidence against the assumed mean. In this setting, the sample mean of 2.2 does not provide convincing support for claiming that the population mean is higher than 2.

So the correct conclusion is:

C. Since this result is not unusual, it is not reasonable to conclude that the population mean is higher than 2.

Why this reasoning works

The logic is based on the idea that unusual outcomes suggest the assumed population mean may be wrong. But a probability of 0.1867 is not small enough to make that claim. We expect outcomes like this to happen fairly often under the model.

If a sample mean falls in a common part of the sampling distribution, it should not be used as strong evidence for a different population mean.

Pitfall alert

Students often confuse “unusual” with “large” in the everyday sense. In statistics, unusual has a precise meaning: probability less than 0.05. Another common error is mixing up the sample mean with the population mean and then concluding that a single observed mean automatically proves the true mean changed. It does not. A probability of 0.1867 says the sample result is plausible under the assumed model, so it is weak evidence at best. Also, when a multiple-choice question asks both for unusualness and for a conclusion, each part must be answered consistently from the same probability threshold.

Try different conditions

If the probability were changed to $P(\bar{x}\ge 2.2)=0.0321$, then the result would be unusual because it would be below 5%. In that case, the conclusion would shift toward saying the sample provides evidence that the assumed mean may be too low. For example, a variant prompt might ask: “If $P(\bar{x}\ge 2.2)=0.0321$, is the result unusual, and what should we conclude about the population mean?” The answer would be that the result is unusual and it may be reasonable to suspect the population mean is greater than 2.

Further reading

sampling distribution, unusual event, population mean