Question
A tyre manufacturer found that the sample mean tread life of 16 radial tyres tested was 60000 km with standard deviation 12000 km. Construct a 90% confidence interval estimate for the true mean tread life of all radial tyres manufactured. Assume that the population tyres manufactured is normal distributed.
Give the upper limit.
Give the upper limit.
Answers
GPT-4o mini
To construct a 90% confidence interval for the true mean tread life of all radial tyres, we can use the formula for the confidence interval which is given by:
\[
\text{CI} = \bar{x} \pm z_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)
\]
Where:
- \(\bar{x}\) = sample mean
- \(s\) = sample standard deviation
- \(n\) = sample size
- \(z_{\alpha/2}\) = z-value that corresponds to the desired confidence level
### Given:
- Sample mean (\(\bar{x}\)) = 60000 km
- Sample standard deviation (\(s\)) = 12000 km
- Sample size (\(n\)) = 16
### Step 1: Find \(z_{\alpha/2}\) for a 90% confidence level
For a 90% confidence interval, we have:
\[
\alpha = 1 - 0.90 = 0.10
\]
\[
\alpha/2 = 0.05
\]
From the z-table, the z-value that corresponds to 0.05 in the upper tail is approximately:
\[
z_{\alpha/2} \approx 1.645
\]
### Step 2: Calculate the standard error (SE)
\[
SE = \frac{s}{\sqrt{n}} = \frac{12000}{\sqrt{16}} = \frac{12000}{4} = 3000
\]
### Step 3: Construct the confidence interval
Now we can calculate the confidence interval:
\[
\text{Margin of error} = z_{\alpha/2} \times SE = 1.645 \times 3000 = 4935
\]
Therefore, the confidence interval is:
\[
\text{CI} = \bar{x} \pm \text{Margin of error} = 60000 \pm 4935
\]
### Step 4: Calculate the upper limit
Upper limit = \(60000 + 4935 = 64935\)
Thus, the **upper limit** of the 90% confidence interval estimate for the true mean tread life of all radial tyres manufactured is:
\[
\boxed{64935} \text{ km}
\]
\[
\text{CI} = \bar{x} \pm z_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)
\]
Where:
- \(\bar{x}\) = sample mean
- \(s\) = sample standard deviation
- \(n\) = sample size
- \(z_{\alpha/2}\) = z-value that corresponds to the desired confidence level
### Given:
- Sample mean (\(\bar{x}\)) = 60000 km
- Sample standard deviation (\(s\)) = 12000 km
- Sample size (\(n\)) = 16
### Step 1: Find \(z_{\alpha/2}\) for a 90% confidence level
For a 90% confidence interval, we have:
\[
\alpha = 1 - 0.90 = 0.10
\]
\[
\alpha/2 = 0.05
\]
From the z-table, the z-value that corresponds to 0.05 in the upper tail is approximately:
\[
z_{\alpha/2} \approx 1.645
\]
### Step 2: Calculate the standard error (SE)
\[
SE = \frac{s}{\sqrt{n}} = \frac{12000}{\sqrt{16}} = \frac{12000}{4} = 3000
\]
### Step 3: Construct the confidence interval
Now we can calculate the confidence interval:
\[
\text{Margin of error} = z_{\alpha/2} \times SE = 1.645 \times 3000 = 4935
\]
Therefore, the confidence interval is:
\[
\text{CI} = \bar{x} \pm \text{Margin of error} = 60000 \pm 4935
\]
### Step 4: Calculate the upper limit
Upper limit = \(60000 + 4935 = 64935\)
Thus, the **upper limit** of the 90% confidence interval estimate for the true mean tread life of all radial tyres manufactured is:
\[
\boxed{64935} \text{ km}
\]
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