Estimation and Hypothesis Testing

ESTIMATION & INFERENCE

Chapter 9, Estimation and Hypothesis Testing , marks the bridge from theoretical probability and the study of sample statistics (Chapters 7 and 8) to the core application of statistical inference. This chapter explores methods used to leverage observed data to make formal statements, educated guesses, or concrete judgments about unknown characteristics of the underlying population distribution.

9.1 The Core Problem: Estimating Unknown Parameters

In statistics, we often assume an observed sample ( $X_1, X_2, \dots, X_n$ ) comes from a distribution with an unknown shape or parameter(s) ( $p_1, p_2, \dots, p_d$ , denoted as $p$ ). The goal of estimation is to use the sample data to provide a best guess for these unknown parameters.

Concept: Point Estimator

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Point Estimator

A point estimator is a function, $g(X_1, X_2, \dots, X_n)$ , that takes the sample values as input and produces a single value (the estimate) intended to approximate the true parameter(s).

Example: If $X$ has an unknown mean $\mu$ , the sample mean ( $\bar{X}$ ) is an unbiased estimator of $\mu$ .

9.2 Method of Moments

The Method of Moments relies on matching sample moments to population moments.

Concept

Sample Moment: $m_k = \frac{1}{n} \sum_{i=1}^{n} X_i^k$
Population Moment: $\mu_k = E[X^k]$ (function of unknown parameters).
Solve: Set $m_k = \mu_k$ for $k=1 \dots d$ and solve for parameters.

Example: Normal Distribution ( $\mu, \sigma^2$ )

$\hat{\mu} = \bar{X}$
$\hat{\sigma}^2 = \frac{1}{n} \sum X_i^2 - (\bar{X})^2$

9.3 Maximum Likelihood Estimate (MLE)

The MLE method determines the parameter ( $\hat{p}$ ) that makes the observed data “most likely”.

Concept: Likelihood Function

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Likelihood Function

The likelihood function, $L(p)$ , is the joint probability of the sample as a function of the parameters:

$L(p; X_1, \dots, X_n) = \prod_{i=1}^{n} f(X_i | p)$

The MLE ( $\hat{p}$ ) comes from maximising this function (or usually $\ln L$ ).

Example: For Bernoulli( $p$ ), the MLE is the sample proportion $\hat{p} = \bar{X}$ .

9.4 Confidence Intervals

A confidence interval provides a range of values within which the true unknown parameter is expected to lie with a specified confidence level ( $\beta$ ).

Construction

Known $\sigma$ (Z-interval): Uses the Standard Normal distribution ( $Z$ ). $Z = \frac{\sqrt{n}(\bar{X} - \mu)}{\sigma}$
Unknown $\sigma$ (T-interval): Uses the t-distribution ( $t_{n-1}$ ) and sample deviation $S$ . $T = \frac{\sqrt{n}(\bar{X} - \mu)}{S}$

Key Difference: T-intervals are typically wider than Z-intervals, reflecting the extra uncertainty of estimating $\sigma$ .

9.5 Hypothesis Testing

Hypothesis testing is a procedure to judge the plausibility of a conjecture about a parameter.

Core Concepts

Null Hypothesis ( $H_0$ ): Baseline statement (e.g., $\mu = 10$ ).
Alternate Hypothesis ( $H_a$ ): Contradiction (e.g., $\mu \neq 10$ ).
Significance Level ( $\alpha$ ): Threshold for rejection (e.g., 0.05).
P-value: Probability of seeing data this extreme if $H_0$ were true.

Standardized Tests

Test Name	Purpose	Test Statistic	Null Hypothesis
Z-test	Test $\mu$ ( $\sigma$ known/ $n$ large)	$Z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}}$	$\mu = \mu_0$
T-test	Test $\mu$ ( $\sigma$ unknown)	$T = \frac{\bar{X} - \mu_0}{S/\sqrt{n}}$	$\mu = \mu_0$
$\chi^2$ -test	Test $\sigma^2$	$W = \frac{(n-1)S^2}{\sigma_0^2}$	$\sigma = \sigma_0$
F-test	Compare two variances	$R = \frac{S_1^2}{S_2^2}$	$\sigma_1 = \sigma_2$

Example: Z-test on Sample Mean

Hypothesis Test

Scenario: Normal population, $\sigma=3.0$ , $n=16$ , $\bar{X}=10.2$ . Test $H_0: \mu = 9.5$ vs $H_a: \mu > 9.5$ at $\alpha=0.05$ .

📝 View Detailed Solution ▼

Test Statistic: $Z_{\text{obs}} = \frac{\sqrt{16}(10.2 - 9.5)}{3.0} \approx 0.933$
P-value: $P(Z \geq 0.933) \approx 0.175$ .
Conclusion: Since $0.175 > 0.05$ , we do not reject $H_0$ .

Goodness of Fit ( $\chi^2$ )

Test if observed categorical data matches expected counts.

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Chi-Square Statistic

$\chi^2 = \sum_{j=1}^{k} \frac{(Y_j - np_j)^2}{np_j}$

Analogy: Estimation is like trying to guess the location of a hidden treasure. Point estimation gives you a single set of coordinates, while a confidence interval gives you a small map. Hypothesis testing is like testing a rumour about the treasure’s location ( $H_0$ ) against the evidence found in the field.