Suppose that the antenna links of woodlice are approximately normally distributed with the mean of 0.25 inches and a standard deviation of 0.05 inches. what proportion of woodlice have antenna lengths that are more than 0.18 inches. Round your answer to at least four decimal places.

To solve this problem, we need to find the proportion of woodlice with antenna lengths greater than 0.18 inches. This can be determined using the **z-score** formula and the standard normal distribution table.

### Step 1: Calculate the z-score
The z-score formula is given by:

\[
z = \frac{X - \mu}{\sigma}
\]

Where:
- \( X = 0.18 \) inches (the value we're interested in)
- \( \mu = 0.25 \) inches (the mean)
- \( \sigma = 0.05 \) inches (the standard deviation)

Substituting the values:

\[
z = \frac{0.18 - 0.25}{0.05} = \frac{-0.07}{0.05} = -1.4
\]

### Step 2: Find the corresponding probability
Using the z-score of \(-1.4\), we look up this value in the standard normal distribution table (or use a calculator) to find the cumulative probability up to \( z = -1.4 \).

The cumulative probability for \( z = -1.4 \) is approximately 0.0808.

### Step 3: Find the proportion of woodlice with antenna lengths more than 0.18 inches
Since we want the proportion greater than 0.18 inches:

\[
P(X > 0.18) = 1 - P(X \leq 0.18) = 1 - 0.0808 = 0.9192
\]

### Final Answer
The proportion of woodlice with antenna lengths greater than 0.18 inches is approximately **0.9192** (rounded to four decimal places).

Would you like more details or have any questions about this problem?

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Here are some related questions you might find useful:

1. What proportion of woodlice have antenna lengths less than 0.22 inches?
2. What is the z-score corresponding to an antenna length of 0.30 inches?
3. What proportion of woodlice have antenna lengths between 0.20 and 0.30 inches?
4. What antenna length corresponds to the 95th percentile?
5. How would the proportion change if the mean antenna length increased to 0.27 inches?
6. What is the probability that a woodlouse has an antenna length within one standard deviation of the mean?
7. What proportion of woodlice have antenna lengths less than the mean?
8. How would the proportion of woodlice with antenna lengths greater than 0.18 inches change if the standard deviation were 0.10 inches?

**Tip:** When working with normal distributions, remember that a z-score tells you how many standard deviations away from the mean a value is.

Calculate the proportion of woodlice with antenna lengths greater than 0.18 inches using the z-score formula and normal distribution table. Learn how to apply statistical concepts like z-scores, mean, and standard deviation in solving this problem.

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