To calculate the 90% confidence interval for the true mean length of the bolt, we need to find the margin of error and then use it to create the interval.
The margin of error is given by:
Margin of error = z* (sigma / sqrt(n))
where z* is the z-score corresponding to the desired confidence level, sigma is the standard deviation, and n is the sample size.
For a 90% confidence level, the z-score is 1.645 (using a standard normal distribution table or calculator).
The standard deviation is the square root of the variance, so sigma = sqrt(0.09) = 0.3.
Substituting these values, we get:
Margin of error = 1.645 * (0.3 / sqrt(9)) = 0.33
To create the confidence interval, we add and subtract the margin of error from the sample mean:
Confidence interval = (sample mean - margin of error, sample mean + margin of error)
Substituting the sample mean and margin of error, we get:
Confidence interval = (3 - 0.33, 3 + 0.33) = (2.67, 3.33)
Therefore, we can say with 90% confidence that the true mean length of the bolt is between 2.67 inches and 3.33 inches.