To construct a 99% confidence interval for the true mean tread life of all radial tyres, we can use the formula for the confidence interval when the sample size is large (which applies here since \( n = 49 \)):
\[ \text{Confidence Interval} = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right) \]
Where:
- \( \bar{x} \) is the sample mean
- \( z \) is the z-value corresponding to the desired confidence level
- \( s \) is the sample standard deviation
- \( n \) is the sample size
Given:
- Sample mean \( \bar{x} = 52345 \) km
- Sample standard deviation \( s = 12943 \) km
- Sample size \( n = 49 \)
- Confidence level = 99%
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Find the z-value for a 99% confidence level. For a 99% confidence interval, we look up the z-value in the standard normal distribution. The z-value for 99% confidence is approximately 2.576.
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Calculate the standard error (SE): \[ SE = \frac{s}{\sqrt{n}} = \frac{12943}{\sqrt{49}} = \frac{12943}{7} \approx 1849 \]
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Calculate the margin of error (ME): \[ ME = z \times SE = 2.576 \times 1849 \approx 4760.424 \]
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Construct the confidence interval:
- Lower limit: \( \bar{x} - ME = 52345 - 4760.424 \approx 47584.576 \)
- Upper limit: \( \bar{x} + ME = 52345 + 4760.424 \approx 57105.424 \)
Thus, the 99% confidence interval for the true mean tread life of all radial tyres is approximately \( (47584.576, 57105.424) \).
To answer your question specifically, the lower limit of the confidence interval is approximately 47584.58 km (rounded to two decimal places).
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