Question

which of the following is a biased estimator?
a. proportion
b. standard deviation
c. variance
d. mean

Explanation:

Step1: Recall the properties of estimators

An estimator is biased if the expected value of the estimator is not equal to the parameter it is estimating.
For a sample, the formula for sample variance is \(s^{2}=\frac{1}{n - 1}\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\) (unbiased estimator of population variance \(\sigma^{2}\)), and the formula for sample standard deviation is \(s=\sqrt{\frac{1}{n}\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\) (biased estimator of population standard deviation \(\sigma\)). The sample mean \(\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}\) is an unbiased estimator of the population mean \(\mu\), and sample proportion \(\hat{p}=\frac{X}{n}\) (where \(X\) is the number of successes in \(n\) trials) is an unbiased estimator of the population proportion \(p\).

Answer:

B. standard deviation