To calculate the margin of error for the average shell diameter of box turtles, we can use the formula for the margin of error (ME) in the context of a confidence interval. The formula is:
\[ ME = z \cdot \left( \frac{s}{\sqrt{n}} \right) \]
Where:
- \( z \) is the z-score corresponding to the desired level of confidence.
- \( s \) is the sample standard deviation.
- \( n \) is the sample size.
For this example, we need to decide on a confidence level to determine the z-score. A common confidence level is 95%, which corresponds to a z-score of approximately 1.96.
Given:
- Sample size, \( n = 54 \)
- Sample mean, \( \bar{x} = 8.1 \) cm (not needed for the margin of error calculation)
- Sample standard deviation, \( s = 2.3 \) cm
Now we can calculate \( \frac{s}{\sqrt{n}} \):
\[ \frac{s}{\sqrt{n}} = \frac{2.3}{\sqrt{54}} \approx \frac{2.3}{7.348} \approx 0.312 \]
Now we can find the margin of error:
\[ ME = z \cdot \left( \frac{s}{\sqrt{n}} \right) = 1.96 \cdot 0.312 \approx 0.612 \]
Finally, rounding to the nearest hundredth:
\[ \text{Margin of error} \approx 0.61 \]
Thus, the sample’s margin of error is 0.61 cm.