Blue Tuna lengths are normally distributed with population mean μ = 70 inches and population standard deviation σ = 11 inches...

the smallest 5% are less than _____ inches.

To find the value below which the smallest 5% of Blue Tuna lengths fall, we need to determine the 5th percentile of a normal distribution. This corresponds to finding a value \( x \) such that 5% of the distribution lies below it.

We use the following relationship for a normally distributed variable \( X \):

\[
P(X < x) = 0.05
\]

Given:
- Population mean \( \mu = 70 \) inches
- Population standard deviation \( \sigma = 11 \) inches

We need to find \( x \) using the z-score formula:

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

From standard normal distribution tables or a calculator, the z-score corresponding to the 5th percentile is approximately \( z = -1.645 \).

Now, solve for \( x \):

\[
-1.645 = \frac{x - 70}{11}
\]

Multiplying both sides by 11:

\[
-1.645 \times 11 = x - 70
\]

\[
-18.095 = x - 70
\]

Adding 70 to both sides:

\[
x = 70 - 18.095 = 51.905
\]

Thus, the smallest 5% of Blue Tuna lengths are less than approximately **51.91 inches**.

Would you like more details or clarifications?

Here are 5 related questions:
1. What is the z-score corresponding to the 95th percentile?
2. How would the result change if the standard deviation was larger?
3. Can you explain how to calculate percentiles using a z-table?
4. What is the probability that a Blue Tuna is longer than 80 inches?
5. How do we find the middle 90% of this distribution?

**Tip:** Percentile values in normal distributions can always be found using z-scores from standard normal tables or statistical calculators.

Blue Tuna lengths are normally distributed with population mean μ = 70 inches and population standard deviation σ = 11 inches. The smallest 5% are less than _____ inches.

Learn how to calculate the 5th percentile of Blue Tuna lengths, which are normally distributed with a mean of 70 inches and a standard deviation of 11 inches. This step-by-step solution demonstrates how to use the z-score formula to find that the smallest 5% of Blue Tuna lengths are less than approximately 51.91 inches.

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