Explanation:

To calculate the 95% confidence interval for the population mean when the population standard deviation is unknown but the sample size is large (n=121), we use the sample standard deviation and the z-distribution.

Step 1: Calculate the standard error of the sample mean $\text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{19}{\sqrt{121}} = \frac{19}{11} = 1.727 \approx 1.73$

Step 2: Determine the z-value for 95% confidence interval For a 95% confidence interval, the z-value (reliability factor) is 1.96.

Step 3: Calculate the margin of error $\text{Margin of Error} = z \times \text{Standard Error} = 1.96 \times 1.73 = 3.3908 \approx 3.40$

Step 4: Calculate the confidence interval

• Lower limit: $45 - 3.40 = 41.60$
• Upper limit: $45 + 3.40 = 48.40$

Therefore, the 95% confidence interval is [41.6 days; 48.4 days].

Why other options are incorrect:

• A [44.7; 45.3]: This interval is too narrow and would correspond to a much smaller standard error.
• C [42.2; 47.8]: This uses an incorrect z-value or standard error calculation.
• D [40.1; 49.8]: This interval is too wide and would correspond to a larger standard error or different confidence level.