Answer :
To find the standard error and sample mean from the confidence interval (1.65, 2.03), we need to interpret this interval in the context of estimating a population parameter (such as the population mean) with a certain level of confidence. This confidence level is generally assumed to be 95%, unless otherwise specified.
1. **Sample Mean (\(\bar{x}\)):**
- The midpoint of the confidence interval is our best estimate of the population mean. In this case, the midpoint is:
\[
\bar{x} = \frac{1.65 + 2.03}{2} = 1.84
\]
- Then, the sample mean (\(\bar{x}\)) is 1.84.
2. **Standard Error (E):**
- The standard error is half the width of the confidence interval. This represents the maximum probable difference between the sample mean and the population mean.
- Interval width = \( 2.03 - 1.65 = 0.38 \)
- Standard error \( E = \frac{\text{Interval width}}{2} = \frac{0.38}{2} = 0.19 \)
Therefore, based on the confidence interval (1.65, 2.03):
- The sample mean (\(\bar{x}\)) is \( \boxed{1.84} \).
- The standard error (E) is \( \boxed{0.19} \).
These values indicate that we estimate that the population mean is approximately 1.84, with a margin of error of approximately 0.19 units.
