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The problem asks whether it is reasonable to assume that insects from an unusual environment have the same probability (0.20) of exhibiting characteristic A as the species in general, given that none of the 18 sampled insects have this characteristic. We are to perform a two-tailed hypothesis test.

### Hypothesis
- **Null Hypothesis (H₀)**: The proportion of insects from the unusual environment that exhibit characteristic A is \( p = 0.20 \).
- **Alternative Hypothesis (H₁)**: The proportion of insects from the unusual environment that exhibit characteristic A is not \( p = 0.20 \).

### Given:
- Population proportion, \( p = 0.20 \)
- Sample size, \( n = 18 \)
- Observed number of insects with characteristic A, \( X = 0 \)
- Sample proportion, \( \hat{p} = \frac{X}{n} = 0 \)

### Test statistic
The test statistic for a proportion is calculated as:
\[
z = \frac{\hat{p} - p}{\sqrt{\frac{p(1 - p)}{n}}}
\]
Substituting the given values:
\[
z = \frac{0 - 0.20}{\sqrt{\frac{0.20(1 - 0.20)}{18}}} = \frac{-0.20}{\sqrt{\frac{0.16}{18}}} = \frac{-0.20}{\sqrt{0.00889}} = \frac{-0.20}{0.0943} \approx -2.12
\]

### P-value
For a two-tailed test, we find the p-value by looking at the standard normal distribution (z-distribution). The z-score of -2.12 corresponds to a left-tail probability of approximately 0.017. Since it’s a two-tailed test, we multiply this by 2 to account for both tails:
\[
\text{P-value} = 2 \times 0.017 = 0.034
\]

### Conclusion
- If we assume a significance level \( \alpha = 0.05 \), the p-value \( 0.034 \) is less than \( \alpha \). Therefore, we reject the null hypothesis.
- This suggests that it is not reasonable to assume that the insects from the unusual environment have the same probability (0.20) of exhibiting characteristic A as the species in general.

Would you like more details or have any questions?  
Here are 5 related questions you might find interesting:
1. How would the conclusion change if the sample size were larger?
2. What is the impact of choosing a different significance level?
3. How do we interpret the test in the context of type I and type II errors?
4. What are some alternative methods for testing proportions in small samples?
5. How would a one-tailed test differ from this two-tailed test?

**Tip:** Always ensure your sample size is adequate for statistical testing to reduce the margin of error and increase the reliability of your results.

It is known that 20% of a certain species of insect exhibit a particular characteristic A. Eighteen insects of that species are obtained from an unusual environment, and none of these have characteristic A. Is it reasonable to assume that insects from that environment have the same probability of 0.20 that the species in general has? Use a two-tailed test.

Perform a two-tailed hypothesis test to determine if insects from an unusual environment have the same probability of exhibiting characteristic A (0.20) as the species in general. Learn how to calculate the z-score and p-value to test the null hypothesis in this detailed statistics problem.

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