Introduction:

In statistical analysis, the confidence interval provides a range of values within which the population parameter is likely to fall. The range is determined by the sample mean and the margin of error, which is influenced by the sample size and the desired level of confidence. In this case, we are given the 95% lower confidence limit (LCL) and upper confidence limit (UCL) for the population mean.

Given Information:

- 95% LCL = 48.04
- 95% UCL = 51.96

Calculating the Margin of Error:

The margin of error (MOE) is the maximum amount by which the sample mean is likely to differ from the population mean. It can be calculated as half the width of the confidence interval.

MOE = (UCL - LCL) / 2

Substituting the given values:

MOE = (51.96 - 48.04) / 2

MOE = 3.92 / 2

MOE = 1.96

Calculating the Confidence Interval:

The confidence interval (CI) can be calculated by adding and subtracting the margin of error from the sample mean.

CI = (sample mean) ± MOE

Since the LCL and UCL represent the lower and upper bounds of the confidence interval, respectively, we can set up the following equations:

LCL = (sample mean) - MOE
UCL = (sample mean) + MOE

Solving for the Sample Mean:

To find the sample mean, we need to calculate the average of the LCL and UCL.

(sample mean) = (LCL + UCL) / 2

Substituting the given values:

(sample mean) = (48.04 + 51.96) / 2

(sample mean) = 100 / 2

(sample mean) = 50

Calculating the Population Standard Deviation:

The population standard deviation (SD) can be estimated using the sample mean and the margin of error. The formula to calculate the sample size required for a given margin of error is:

n = (Z * SD / MOE)^2

Where:
- n is the sample size
- Z is the Z-score corresponding to the desired level of confidence (for 95% confidence level, Z = 1.96)
- SD is the population standard deviation
- MOE is the margin of error

Rearranging the formula to solve for SD:

SD = (n * MOE) / Z

Substituting the given values:

SD = (100 * 1.96) / 1.96

SD = 100

Conclusion:

The value of the population standard deviation when the sample size is 100 and the 95% LCL and UCL are 48.04 and 51.96, respectively, is 100.