To determine the 90% confidence interval for the true mean length of the bolt in the given manufacturing process, we need to follow these steps:
### Step 1: Identify the given information
- Sample size ([tex]\( n \)[/tex]): 9
- Sample mean ([tex]\( \bar{x} \)[/tex]): 3 inches
- Sample variance ([tex]\( s^2 \)[/tex]): 0.09
- Confidence level: 90%
### Step 2: Calculate the sample standard deviation
Since variance ([tex]\( s^2 \)[/tex]) is given, we take the square root to find the sample standard deviation ([tex]\( s \)[/tex]):
[tex]\[ s = \sqrt{0.09} = 0.3 \][/tex]
### Step 3: Find the critical value using the t-distribution
The confidence level is 90%, so the significance level ([tex]\( \alpha \)[/tex]) is:
[tex]\[ \alpha = 1 - 0.90 = 0.10 \][/tex]
Since we are dealing with a two-tailed distribution, we divide [tex]\( \alpha \)[/tex] by 2:
[tex]\[ \frac{\alpha}{2} = 0.05 \][/tex]
With 8 degrees of freedom (df = n - 1 = 9 - 1 = 8), we find the critical t-value ([tex]\( t_{\frac{\alpha}{2}, df} \)[/tex]) from the t-distribution table:
[tex]\[ t_{\frac{\alpha}{2}, 8} = 1.8595 \][/tex]
### Step 4: Calculate the margin of error
The margin of error (ME) is given by the formula:
[tex]\[ \text{ME} = t_{\frac{\alpha}{2}, df} \times \left( \frac{s}{\sqrt{n}} \right) \][/tex]
Substituting the values:
[tex]\[ \text{ME} = 1.8595 \times \left( \frac{0.3}{\sqrt{9}} \right) = 1.8595 \times 0.1 = 0.186 \][/tex]
### Step 5: Determine the confidence interval
To find the confidence interval, we add and subtract the margin of error from the sample mean ([tex]\( \bar{x} \)[/tex]):
- Lower bound:
[tex]\[ \bar{x} - \text{ME} = 3 - 0.186 = 2.814 \][/tex]
- Upper bound:
[tex]\[ \bar{x} + \text{ME} = 3 + 0.186 = 3.186 \][/tex]
### Conclusion
The 90% confidence interval for the true mean length of the bolt is (2.814, 3.186).
From the given options, the correct interval is:
[tex]\[ 2.8140 \text{ to } 3.1860 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{2.8140 \text{ to } 3.1860} \][/tex]
