Part a
State the main assumptions that must be verified before conducting a T test and a Z test for comparing a sample mean to a hypothesized population mean.
Comparison of Sample Variance Estimators
Consider a random sample \(X_1, X_2, \ldots, X_n\) drawn from a distribution with true variance \(\sigma^2\). The usual sample variance is defined as \(s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i – \bar{X})^2\), which is an unbiased estimator of \(\sigma^2\). An alternative estimator uses the denominator \(n\) instead, i.e., \(\tilde{s}^2 = \frac{1}{n}\sum_{i=1}^{n}(X_i – \bar{X})^2\). (a) Derive the expected value of […]
Derivation of the Unbiased Sample Variance Estimator
In large-scale data analysis, the sample variance is a key statistic used to estimate the population variance. A common, but biased, estimator for sample variance is given by \( s^2 = \frac{1}{n} \sum_{i=1}^{n}(x_i – \bar{x})^2 \). (a) Explain mathematically why this naive estimator is biased when estimating the population variance. (b) Derive the formula for […]
Derivation of the Unbiased Sample Variance
Let \(X_1, X_2, \ldots, X_n\) be a random sample from a normal distribution with mean \(\mu\) and variance \(\sigma^2\). Show that the sample variance \(S^2 = \frac{1}{n-1}\sum_{i=1}^{n} (X_i-\overline{X})^2\) is an unbiased estimator of \(\sigma^2\), that is, prove that \(E(S^2) = \sigma^2\). Use the decomposition \(\sum_{i=1}^{n}(X_i-\mu)^2 = \sum_{i=1}^{n}(X_i-\overline{X})^2 + n(\overline{X}-\mu)^2\) and the properties of the normal […]
Deriving the Unbiased Estimator for Population Variance
Consider a random sample \(X_1, X_2, \dots, X_n\) drawn from a population with true variance \(\sigma^2\). The sample variance is defined as \(s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i – \bar{X})^2\). Derive the expected value \(E(s^2)\) and show that \(s^2\) is an unbiased estimator of \(\sigma^2\).
Derivation of Unbiased Sample Variance
Consider a random sample \( x_1, x_2, \ldots, x_n \) drawn from a population with mean \(\mu\) and variance \(\sigma^2\). The sample mean is given by \(\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i\), and the sample variance is defined as \(S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2\). Derive why the denominator is \(n-1\) by showing that \(E(S^2)=\sigma^2\). In your answer, explain the role of degrees […]
Deriving the Sample Variance from a Data Set
Consider the following dataset of four values: 4, 8, 6, 10. (a) Calculate the sample mean \(\overline{x}\). (b) Using the sample variance formula \(s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\overline{x})^2\), compute the sample variance step-by-step. Show all steps in your workings.