A tyre manufacturer found that the sample mean tread life of 49 radial tyres tested was 52345 km with standard deviation 12943 km. Construct a 99% confidence interval estimate for the true mean tread life of all radial tyres manufactured.

Give the upper limit

1 answer

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:

\[ \boxed{57095.8 \text{ km}} \]