Outliers are observations that stand too different from others in a set of observations. When present in a data set, they affect both descriptive and inferential statistics. This work therefore, studies the sensitivity and robustness of one sample test statistics to outliers so as to know the appropriate one to test hypothesis about the population parameter when outliers are present. One sample test statistics considered are: parametrics test (Student t-test and ztest), non-parametric test (Wilcoxon Sign test (Distribution Sign test (DST), Asymptotic Sign test (AST)), Wilcoxon Signed rank test (Distribution Wilcoxon Signed rank test (DWST) and Asymptotic (AWST)), t-test for rank transformation (Rt-test) and Trimmed t-test statistics (Tt-test). Monte Carlo experiments, replicated five thousand (5000) times, were conducted at eight (8) sample sizes (10, 15, 20, 25, 30, 35, 40 and 50) by simulating data from normal distribution. At each of the sample sizes, 10% and 20% of the generated data were randomly selected and invoked with various magnitude of outliers (-10, -9, -8,… 8, 9, 10). The test statistics were compared at three levels of significance, 0.1, 0.05 and 0.01. A test is considered robust if its estimated error rate approximates the true error rate and has the highest number of times it approximates the error rate when counted over the percentage (%) of outliers, magnitudes of outliers and levels of significance; and if the counts is minimum the test statistics is sensitive. At all the three (3) levels of significance, results revealed that Type 1 error rates of Student t-test, Rt-test and AWST statistics are good; and that z-test and Student t-test statistics are most sensitive to outliers. The statistics robustness is affected by the levels of significance in that the sign test (DST and AST) is robust at 0.1; Tt-test and Wilcoxon Sign Rank test (DWST and AWST) at 0.05; and DST, AWST, Tttest and AST at 0.01 level of significance. Consequently, the Sign test