Question
A tyre manufacturer found that the sample mean tread life of 20 radial tyres tested was 50000 km. It is also known that the population standard deviation is 12000 km. Construct a 95% confidence interval estimate for the true mean tread life of all radial tyres manufactured. Assume that the population tyres manufactured is normal distributed.
6.1 Give the table value.
6.2 Give the lower limit.
6.3 Give the upper limit
6.1 Give the table value.
6.2 Give the lower limit.
6.3 Give the upper limit
Answers
GPT-4o mini
To construct a 95% confidence interval for the true mean tread life of all radial tyres manufactured, we will use the following formula for the confidence interval:
\[
\text{CI} = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)
\]
where:
- \(\bar{x}\) = sample mean
- \(Z\) = Z-value corresponding to the desired confidence level
- \(\sigma\) = population standard deviation
- \(n\) = sample size
Given data:
- Sample mean (\(\bar{x}\)) = 50000 km
- Population standard deviation (\(\sigma\)) = 12000 km
- Sample size (\(n\)) = 20
- Confidence level = 95%
### 6.1 Give the table value.
For a 95% confidence level, the critical Z-value can be found using Z-tables or standard normal distribution calculators. The Z-value for a 95% confidence level (two-tailed) is approximately:
\[
Z \approx 1.96
\]
### 6.2 Give the lower limit.
Now, calculate the standard error (SE):
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{12000}{\sqrt{20}} \approx \frac{12000}{4.472} \approx 2683.28 \text{ km}
\]
Using this information, we can calculate the lower limit (LL) of the confidence interval:
\[
LL = \bar{x} - Z \times SE = 50000 - 1.96 \times 2683.28 \approx 50000 - 5267.24 \approx 44732.76 \text{ km}
\]
### 6.3 Give the upper limit.
Now calculate the upper limit (UL) of the confidence interval:
\[
UL = \bar{x} + Z \times SE = 50000 + 1.96 \times 2683.28 \approx 50000 + 5267.24 \approx 55267.24 \text{ km}
\]
### Summary
- **Table value (Z)**: 1.96
- **Lower limit**: 44732.76 km
- **Upper limit**: 55267.24 km
Thus, the 95% confidence interval for the true mean tread life of all radial tyres manufactured is approximately (44732.76 km, 55267.24 km).
\[
\text{CI} = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)
\]
where:
- \(\bar{x}\) = sample mean
- \(Z\) = Z-value corresponding to the desired confidence level
- \(\sigma\) = population standard deviation
- \(n\) = sample size
Given data:
- Sample mean (\(\bar{x}\)) = 50000 km
- Population standard deviation (\(\sigma\)) = 12000 km
- Sample size (\(n\)) = 20
- Confidence level = 95%
### 6.1 Give the table value.
For a 95% confidence level, the critical Z-value can be found using Z-tables or standard normal distribution calculators. The Z-value for a 95% confidence level (two-tailed) is approximately:
\[
Z \approx 1.96
\]
### 6.2 Give the lower limit.
Now, calculate the standard error (SE):
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{12000}{\sqrt{20}} \approx \frac{12000}{4.472} \approx 2683.28 \text{ km}
\]
Using this information, we can calculate the lower limit (LL) of the confidence interval:
\[
LL = \bar{x} - Z \times SE = 50000 - 1.96 \times 2683.28 \approx 50000 - 5267.24 \approx 44732.76 \text{ km}
\]
### 6.3 Give the upper limit.
Now calculate the upper limit (UL) of the confidence interval:
\[
UL = \bar{x} + Z \times SE = 50000 + 1.96 \times 2683.28 \approx 50000 + 5267.24 \approx 55267.24 \text{ km}
\]
### Summary
- **Table value (Z)**: 1.96
- **Lower limit**: 44732.76 km
- **Upper limit**: 55267.24 km
Thus, the 95% confidence interval for the true mean tread life of all radial tyres manufactured is approximately (44732.76 km, 55267.24 km).