The difference between B and C is in the choice of critical value. A typical symmetric confidence interval for a location parameter has the form
where in turn
The choice of critical value is informed by the nature of the sampling distribution. When a normally distributed population has unknown variance, the sampling distribution of the mean is Student's t-distributed; however, when the sample size is large, the difference in critical values is negligible; i.e.,
where is the upper quantile of the Student's t distribution with degrees of freedom, and is the upper quantile of the standard normal distribution. So for scenario B, you will use for a two-sided CI with large sample size as an approximation; for scenario C, you will use for the critical value.
The difference between A and B lies in the standard deviation. In scenario A, it is presumed to be known, thus the standard error will use the population standard deviation in the numerator. In scenario B, it is unknown, thus you will use the unbiased estimator of the standard deviation
In practice, the factor of rather than n makes little difference when n is large as in this case. But in scenario C, as is also unknown and must be estimated from the sample, you must use , otherwise your CI will not have the required coverage probability even if you correctly use the t-distribution quantile for the critical value.
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