To calculate the margin of error (ME) for the sample mean, you can use the formula:
\[ ME = z \times \left( \frac{s}{\sqrt{n}} \right) \]
where:
- \( z \) is the z-value corresponding to the desired confidence level,
- \( s \) is the sample standard deviation,
- \( n \) is the sample size.
Assuming a common confidence level of 95%, the z-value is approximately 1.96.
Given:
- \( s = 2.3 \) cm
- \( n = 54 \)
Now, we can compute the margin of error.
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Calculate the standard error (SE): \[ SE = \frac{s}{\sqrt{n}} = \frac{2.3}{\sqrt{54}} \] First, calculate \( \sqrt{54} \): \[ \sqrt{54} \approx 7.348 \] Now compute the SE: \[ SE = \frac{2.3}{7.348} \approx 0.313 \]
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Now, calculate the margin of error: \[ ME = 1.96 \times 0.313 \approx 0.613 \]
Finally, round the margin of error to the nearest hundredth: \[ ME \approx 0.61 , \text{cm} \]
Thus, the sample’s margin of error is approximately 0.61 cm.