Given:
- 95% LCL = 48.04
- 95% UCL = 51.96
- Sample size (n) = 100

To find: Population variance

Approach:
The formula to calculate the confidence interval for population mean when the sample size (n) is greater than 30 is:

Confidence Interval (CI) = Sample Mean ± Zα/2 * (Standard Error)

Where
- Zα/2 is the Z value for the level of confidence α/2
- Standard Error = Population Standard Deviation / sqrt(n)

As the population variance is unknown, we will use the sample standard deviation as an estimate of it.

Formula to calculate the sample standard deviation (s):
s = sqrt [ Σ(xi - x̄)2 / (n - 1) ]

Where
- xi is the i-th observation
- x̄ is the sample mean

Once we have the sample standard deviation, we can calculate the standard error and then use the confidence interval formula to find the population mean.

Steps:
1. Calculate the sample mean (x̄):
x̄ = (LCL + UCL) / 2
= (48.04 + 51.96) / 2
= 50

2. Calculate the Z value:
For 95% confidence interval, α = 0.05 and α/2 = 0.025
Using a Z table or calculator, we can find the Z value for 0.025 = 1.96

3. Calculate the sample standard deviation (s):
We do not have the individual observations, so we cannot calculate the sample standard deviation.

4. Calculate the standard error:
Standard Error = s / sqrt(n)
= s / sqrt(100)
= s / 10

5. Use the confidence interval formula to find the population mean:
CI = Sample Mean ± Zα/2 * (Standard Error)
= 50 ± 1.96 * (s / 10)

We know that the confidence interval is (48.04, 51.96), so we can write two equations using the above formula and solve for s:

48.04 = 50 - 1.96 * (s / 10)
51.96 = 50 + 1.96 * (s / 10)

Solving the above equations, we get:
s = 1

6. Calculate the population variance:
Population Variance = s2 * (n - 1)
= 1^2 * (100 - 1)
= 99

Therefore, the value of the population variance when the sample size is 100 is 10 (Option B).