To construct a 95% confidence interval for the mean height of all Hereford calves at birth, we can use the formula for the confidence interval based on the sample mean and sample standard deviation.
The formula for the confidence interval is given by:
\[ \text{Confidence Interval} = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right) \]
where:
- \( \bar{x} \) is the sample mean,
- \( z \) is the z-score corresponding to the desired confidence level,
- \( s \) is the sample standard deviation,
- \( n \) is the sample size.
Given Data:
- \( \bar{x} = 71 , \text{cm} \)
- \( s = 6.7 , \text{cm} \)
- \( n = 50 \)
Step 1: Find the z-score for a 95% confidence level
For a 95% confidence interval, the z-score is approximately 1.96 (you can find this using a z-table or a normal distribution calculator).
Step 2: Calculate the standard error (SE)
\[ SE = \frac{s}{\sqrt{n}} = \frac{6.7}{\sqrt{50}} \approx \frac{6.7}{7.071} \approx 0.9487 \]
Step 3: Calculate the margin of error (ME)
\[ ME = z \cdot SE = 1.96 \cdot 0.9487 \approx 1.858 \]
Step 4: Construct the confidence interval
Now we can calculate the confidence interval: \[ \text{Confidence Interval} = \bar{x} \pm ME = 71 \pm 1.858 \]
This gives us: \[ (71 - 1.858, 71 + 1.858) = (69.142, 72.858) \]
Conclusion
The 95% confidence interval for the mean height of all Hereford calves at birth is approximately \( (69.14 , \text{cm}, 72.86 , \text{cm}) \).
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