In surveys, one may face the problem of influential units at the estimation stage. A unit is said to be influential if its inclusion or exclusion from the sample has a drastic impact on the estimates. This is a common situation in business surveys as the distribution of economic variables tends to be highly skewed. We study and examine some commonly used estimators and predictors of a population total and propose a robust estimator and predictor based on an adaptive tuning constant. The proposed tuning constant is based on the concept of conditional bias of a unit, which is a measure of influence. We present the results of a simulation study that compares the performance of several estimators and predictors in terms of bias and efficiency.