Three different statistics are being considered for estimating a population characteristic. The sampling distributions of the three statistics are shown in the following illustration. Statistic II Statistic III Statistic I Value of population characteristic Which of these statistics are unbiased estimators of the population characteristic? (Select all that apply.) Statistic I O Statistic II O Statistic III XGreat let's take a closer, step-by-step look at the image to correctly identify which statistics are unbiased estimators. __ Step-by-Step Analysis |) Step 1: What is an Unbiased Estimator? An unbiased estimator is one whose sampling distribution has its mean (center) exactly at the true population value. e Visually, this means the peak (center) of the curve should align with the true population characteristic, as shown on the horizontal axis. a) Step 2: Examine the Sampling Distributions in the Image e The value of the population characteristic is marked by a vertical line in the center of the graph. e There are three bell-shaped curves representing the sampling distributions of: Statistic | (left curve, wider) * Statistic Il (center curve, narrow and peaked) e Statistic Ill (right curve) Let's determine where each distribution is centered relative to the population value: } Statistic |: e The peak of the curve aligns with the population characteristic. | Unbiased Statistic Il: e Also centered exactly at the population characteristic. e |. Unbiased } Statistic III: e The curve is shifted to the right of the population characteristic. e Biased Step 3: Interpretation Only Statistic | and Statistic Il are centered at the true population value, so they are unbiased. Statistic Ill is not centered at the population value, so it is biased. | Final Correct Answer: @Statistic! Qj Statistic ll e [ | Statistic Ill | Correct unbiased estimators: Statistic | and Statistic Il Original Selection in Image: The image shows only Statistic | was selected, which is incorrect by itself. That's why a red 9 is shown