Study Notes: Hypothesis Testing for Two Means, Inferences for Proportions, and Chi-Square Procedures

Pooled t-Test

The pooled t-test is used to compare the means of two independent populations when the population variances are assumed to be equal.

• Assumptions: Both populations are normally distributed, and the variances are equal.

• Test Statistic: The pooled t-test statistic is calculated as:

$t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$ where $s_p$ is the pooled standard deviation:

$s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$

• Degrees of Freedom: $df = n_1 + n_2 - 2$

• Comparing the average test scores of two classes with similar variances.

Non-Pooled t-Test

The non-pooled (Welch's) t-test is used when the variances of the two populations are not assumed to be equal.

• Test Statistic:

$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Degrees of Freedom: Calculated using the Welch-Satterthwaite equation:

$df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}$

• Comparing the average heights of two groups with different variances.

Paired t-Test

The paired t-test is used to compare the means of two related groups (e.g., before-and-after measurements on the same subjects).

• Test Statistic:

$t = \frac{\bar{d}}{s_d / \sqrt{n}}$ where $\bar{d}$ is the mean of the differences and $s_d$ is the standard deviation of the differences.

• Degrees of Freedom: $df = n - 1$

• Example: Measuring blood pressure before and after treatment in the same patients.

Estimating an Unknown Population Proportion (Sample Proportion)

Population proportion estimation involves using a sample proportion to estimate the true proportion in the population.

• Sample Proportion: $\hat{p} = \frac{x}{n}$, where $x$ is the number of successes and $n$ is the sample size.

Sampling Distribution for Sample Proportion

The sampling distribution of the sample proportion describes the distribution of $\hat{p}$ over many samples.

• Mean: $\mu_{\hat{p}} = p$

• Standard Deviation (Standard Error): $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

• Approximate Normality: For large $n$, $\hat{p}$ is approximately normal if $np \geq 10$ and $n(1-p) \geq 10$.

1-Proportion Z-Interval Procedure (Confidence Interval)

This procedure is used to construct a confidence interval for a population proportion.

• Formula:

$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Where: $z^*$ is the critical value from the standard normal distribution for the desired confidence level.

• Estimating the proportion of voters who support a candidate.

Margin of Error

The margin of error quantifies the maximum expected difference between the sample proportion and the true population proportion at a given confidence level.

• Calculation:

$E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Finding Sample Size: To achieve a margin of error less than a specified value $E$:

$n = \frac{z^{*2} \hat{p}(1-\hat{p})}{E^2}$

• Determining how many people to survey to estimate a proportion within 3% margin of error.

Hypothesis Tests for One Population Proportion

Used to test claims about a population proportion.

• Test Statistic:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• Where: $p_0$ is the hypothesized population proportion.

• Testing if the proportion of defective items is greater than a specified value.

Chapter 13 - Chi-Square Procedures

Chi-Square Distribution and Degrees of Freedom

The chi-square distribution is used in tests of categorical data, such as goodness-of-fit and independence tests.

• Degrees of Freedom (df): For a goodness-of-fit test, $df = k - 1$, where $k$ is the number of categories. For a contingency table, $df = (r-1)(c-1)$, where $r$ is the number of rows and $c$ is the number of columns.

The Chi-Square Distribution Table

The chi-square table provides critical values for the chi-square statistic at various significance levels and degrees of freedom.

• Purpose: Used to determine the rejection region for hypothesis tests involving the chi-square statistic.

Chi-Square Goodness of Fit Test

This test determines whether observed categorical data fit a specified distribution.

• Test Statistic:

$\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}$ where $O_i$ is the observed frequency and $E_i$ is the expected frequency for category $i$.

• Critical Value Approach: Compare the test statistic to the critical value from the chi-square table.

• P-Value Approach: Find the probability of observing a value as extreme as the test statistic.

• Testing if a die is fair.

Chi-Square Independence Test

This test assesses whether two categorical variables are independent.

• Contingency Tables (Two-Way Tables): Used to organize observed frequencies for two categorical variables.

• Expected Cell Values:

$E_{ij} = \frac{(\text{row total})_i \times (\text{column total})_j}{\text{grand total}}$

• Test Statistic:

$\chi^2 = \sum_{i=1}^r \sum_{j=1}^c \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$

• Critical Value Approach: Compare the test statistic to the critical value for the appropriate degrees of freedom.

• P-Value Approach: Calculate the probability of observing a test statistic as extreme as the one computed.

• Testing if gender and voting preference are independent.

Summary Table: Chi-Square Test Types

Test	Purpose	Degrees of Freedom	Example
Goodness of Fit	Test if observed frequencies match expected distribution	$k - 1$	Testing if a die is fair
Test of Independence	Test if two categorical variables are independent	$(r-1)(c-1)$	Testing if gender and voting preference are independent