Question

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

Answer 1

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:**

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