Final answer:
To construct the 90% confidence interval for the population mean operational time before failure, the sample mean is calculated to be approximately 349.33 hours. With a Z-value of 1.645 for a 90% confidence level and a known population standard deviation, the margin of error is found and the interval is approximately (329.59, 369.07) hours.
Step-by-step explanation:
To construct a 90% confidence interval for the population mean operational time before failure using the provided sample data, we use the formula for the confidence interval for a population mean when the population standard deviation is known:
CI = μ ± (Z * (σ / √n))
μ is the sample mean.
Z is the Z-value from the standard normal distribution for the given confidence level.
σ is the population standard deviation.
n is the sample size.
First, we calculate the sample mean (μ). Add up all the sample values and divide by the number of samples (n).
Sample mean: μ = (322 + 350 + 346 + 347 + 335 + 323 + 341 + 355 + 329) / 9 = 3144 / 9 ≈ 349.33
Next, we look up the Z-value for a 90% confidence level, which is 1.645 for one-tailed (or 90/2 = 45% for two-tailed).
Now, we can calculate the margin of error:
Margin of error: E = Z * (σ / √n) = 1.645 * (36 / √9) ≈ 1.645 * 12 ≈ 19.74
Finally, we construct the confidence interval:
CI = 349.33 ± 19.74 ≈ (329.59, 369.07)
Thus, the 90% confidence interval for the population mean operational time before failure is approximately (329.59, 369.07) hours.