We study the utility maximization problem of a portfolio of one risky asset, a stock, and one riskless asset, a bond, under Knightian uncertainty on the drift term representing the long term growth rate of the risky asset. We further assume that the investor has a prior estimate about the drift term, so that we incorporate into the model a penalty term for deviating from the prior about the mean. We provide explicit solutions, when the investor has logarithmic, power and exponential utility functions. A numerical case study with respect to different utilities has also been presented.