To construct a 99% confidence interval for the true mean tread life of all radial tyres, we can use the formula for a confidence interval:
\[ \text{Confidence interval} = \bar{x} \pm z \left(\frac{s}{\sqrt{n}}\right) \]
where:
- \(\bar{x}\) = sample mean
- \(z\) = z-score for the desired confidence level
- \(s\) = sample standard deviation
- \(n\) = sample size
Given data:
- Sample mean (\(\bar{x}\)) = 52345 km
- Sample standard deviation (\(s\)) = 12943 km
- Sample size (\(n\)) = 49
Step 1: Find the z-score for a 99% confidence level.
The z-score for a 99% confidence level is approximately 2.576 (you can find this value in a z-table or using a standard normal distribution calculator).
Step 2: Calculate the standard error (SE):
\[ SE = \frac{s}{\sqrt{n}} = \frac{12943}{\sqrt{49}} = \frac{12943}{7} \approx 1849.0 \]
Step 3: Calculate the margin of error (ME):
\[ ME = z \cdot SE = 2.576 \cdot 1849.0 \approx 4750.8 \]
Step 4: Determine the confidence interval:
- Lower limit: \(\bar{x} - ME = 52345 - 4750.8 \approx 47594.2\)
- Upper limit: \(\bar{x} + ME = 52345 + 4750.8 \approx 57095.8\)
Therefore, the upper limit of the 99% confidence interval for the true mean tread life of all radial tyres manufactured is approximately:
\[ \boxed{57095.8 \text{ km}} \]