Question 7.4: Radial Tyres Tread Life Study A tyre manufacturer found that......
Radial Tyres Tread Life Study
A tyre manufacturer found that the sample mean tread life of 81 radial tyres tested was 52 345 km. The population standard deviation of radial tyre tread life is 4 016 km and is assumed to be normally distributed.
Estimate, with 99% confidence, the true mean tread life of all radial tyres manufactured. Also interpret the results.
Given \bar{x} = 52 345 km, \sigma = 4 016 km and n = 81 radial tyres.
- Find the standard error of the sample mean.
\sigma_{\bar{x}}=\frac{\sigma }{\sqrt{n} }=\frac{4016}{\sqrt{81} } = 446.22 km
- From the z-table, the 99% confidence level equates to z-limits of ±2.58.
- Compute the margin of error = 2.58 × 446.22 = 1151.25
- Lower limit: 52 345 – 2.58(446.22) = 52 345 – 1 151.25 = 51 193.75 km
Upper limit: 52 345 + 2.58(446.22) = 52 345 + 1 151.25 = 53 496.25 km
Thus the 99% confidence interval is defined as 51 193.75 ≤ μ ≤ 53 496.25 km.
Management Interpretation
The tyre manufacturer can be 99% confident that the actual mean tread life of all their radial tyres is likely to lie between 51 193.75 and 53 496.25 km.