A research scientist wants to know how many times per hour a certain strand of bacteria reproduces. The mean is found to be 10.9 reproductions and the population standard deviation is known to be 2.3 . If a sample of 540 was used for the study, construct the \( 90 \% \) confidence interval for the true mean number of reproductions per hour for the bacteria. Round your answers to one decimal place.

To construct a \(90\%\) confidence interval for the true mean number of reproductions per hour, we can use the formula for the confidence interval: \[ CI = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right) \] Where: - \(\bar{x} = 10.9\) (sample mean) - \(\sigma = 2.3\) (population standard deviation) - \(n = 540\) (sample size) - \(z\) is the z-value for a \(90\%\) confidence level, which is approximately \(1.645\). First, we calculate the standard error (SE): \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{2.3}{\sqrt{540}} \approx \frac{2.3}{23.236} \approx 0.09 \] Next, we calculate the margin of error (ME): \[ ME = z \times SE \approx 1.645 \times 0.09 \approx 0.148 \] Now, we can construct the confidence interval: \[ CI = 10.9 \pm 0.148 \] \[ CI = (10.9 - 0.148, 10.9 + 0.148) = (10.752, 11.048) \] Rounding to one decimal place, the \(90\%\) confidence interval for the true mean number of reproductions per hour for the bacteria is: \[ (10.8, 11.0) \]