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# Statistical Inference ## Point Estimation ### Definition 7.1 A point estimator is a statistic that is used to estimate a population parameter. ### Definition 7.2 Let $W$ be a sample statistic used to estimate the population parameter $\theta$. We define the following: * The estimate is the...

# Statistical Inference ## Point Estimation ### Definition 7.1 A point estimator is a statistic that is used to estimate a population parameter. ### Definition 7.2 Let $W$ be a sample statistic used to estimate the population parameter $\theta$. We define the following: * The estimate is the specific value of the estimator obtained from a sample. * The bias of $W$ is $B(W) = E(W) - \theta$. * If $B(W) = 0$, then $W$ is an unbiased estimator of $\theta$. ### Definition 7.3 The variance of an estimator $W$ is defined as $V(W) = E[W - E(W)]^2$. ### Definition 7.4 The mean squared error (MSE) of an estimator $W$ is defined as $MSE(W) = E[(W - \theta)^2] = V(W) + [B(W)]^2$ ### Definition 7.5 Let $W_1$ and $W_2$ be two unbiased estimators of $\theta$. We say that $W_1$ is more efficient than $W_2$ if $V(W_1) < V(W_2)$. The relative efficiency of $W_1$ with respect to $W_2$ is $\frac{V(W_2)}{V(W_1)}$ ### Theorem 7.1 Let $X_1, X_2,..., X_n$ be a random sample from a population with mean $\mu$ and variance $\sigma^2$. Then, * $\bar{X}$ is an unbiased estimator of $\mu$, and $V(\bar{X}) = \frac{\sigma^2}{n}$ * $S^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n-1}$ is an unbiased estimator of $\sigma^2$. ### Definition 7.6 Let $X_1, X_2,..., X_n$ be a random sample from a probability distribution with parameter $\theta$. The likelihood function, $L(\theta)$, is defined as $L(\theta) = f(x_1, x_2,..., x_n; \theta) = \prod_{i=1}^{n} f(x_i; \theta)$ ### Definition 7.7 Let $\hat{\theta}$ be the value of $\theta$ that maximizes the likelihood function $L(\theta)$. Then $\hat{\theta}$ is called the maximum likelihood estimator (MLE) of $\theta$. ## Interval Estimation ### Definition 8.1 A confidence interval for a parameter $\theta$ is an interval $(L, U)$ such that $P(L < \theta < U) = 1 - \alpha$, where $0 < \alpha < 1$. The quantity $1 - \alpha$ is called the confidence coefficient. ### Definition 8.2 A pivotal quantity is a function of the sample and the parameter that has a distribution that does not depend on the parameter. ### Theorem 8.1 Let $X_1, X_2,..., X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Then, * $\frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \sim N(0, 1)$ * $\frac{\bar{X} - \mu}{S / \sqrt{n}} \sim t_{n-1}$ * $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$ ### Large-Sample Confidence Interval for $\mu$ $\bar{X} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$ ### Small-Sample Confidence Interval for $\mu$ ( $\sigma$ unknown) $\bar{X} \pm t_{\alpha/2, n-1} \frac{S}{\sqrt{n}}$ ## Hypothesis Testing ### Definition 9.1 A statistical hypothesis is a statement about the parameters of a population distribution. ### Definition 9.2 A null hypothesis, denoted by $H_0$, is a statement about the parameters of a population distribution that we want to test. ### Definition 9.3 An alternative hypothesis, denoted by $H_1$, is a statement about the parameters of a population distribution that we will accept if the null hypothesis is rejected. ### Definition 9.4 A test statistic is a statistic that is used to determine whether to reject the null hypothesis. ### Definition 9.5 The rejection region is the set of values of the test statistic for which the null hypothesis is rejected. ### Definition 9.6 A Type I error occurs when we reject the null hypothesis when it is true. The probability of a Type I error is denoted by $\alpha$. ### Definition 9.7 A Type II error occurs when we fail to reject the null hypothesis when it is false. The probability of a Type II error is denoted by $\beta$. ### Definition 9.8 The power of a test is the probability of rejecting the null hypothesis when it is false. The power of a test is $1 - \beta$. ### Definition 9.9 The $p$-value is the probability of observing a test statistic as extreme as, or more extreme than, the one observed, assuming that the null hypothesis is true. ### Large-Sample Test Statistic for $\mu$ $Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}$ ### Small-Sample Test Statistic for $\mu$ ( $\sigma$ unknown) $T = \frac{\bar{X} - \mu_0}{S / \sqrt{n}}$