Hypothesis Testing and Confidence Intervals: Formula Sheet Study Guide

Testing Means & Proportions

Overview

Statistical tests for means and proportions are fundamental tools for making inferences about populations based on sample data. These procedures allow us to test hypotheses and construct confidence intervals for population parameters.

• Estimate: The sample statistic used to estimate the population parameter (e.g., sample mean \( \bar{x} \), sample proportion \( \hat{p} \)).

• Standard Deviation / Standard Error: Measures variability of the estimate.

• Null Hypothesis (H0): The default assumption about the population parameter.

• Test Statistic: Quantifies the difference between the sample estimate and the hypothesized value, standardized by variability.

One-Proportion z-Test

• Estimate: \( \hat{p} = \frac{x}{n} \), \( \hat{q} = 1 - \hat{p} \)

• Standard Error: $\sqrt{\frac{\hat{p}\hat{q}}{n}}$

• Null Hypothesis: \( H_0: p = p_0 \)

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

• Assumptions: Independent observations, SRS, sample size conditions.

• Example: Testing if the proportion of defective items is equal to 0.05.

Two-Proportion z-Test (Unpooled and Pooled)

• Estimate: \( \hat{p}_1 = \frac{x_1}{n_1} \), \( \hat{p}_2 = \frac{x_2}{n_2} \)

• Standard Error (Unpooled): $\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• Null Hypothesis: \( H_0: p_1 - p_2 = p_0 \)

• Test Statistic (Unpooled): $z = \frac{\hat{p}_1 - \hat{p}_2 - p_0}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}$

• Standard Error (Pooled): $\sqrt{\bar{p}(1-\bar{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$ where $\bar{p} = \frac{x_1 + x_2}{n_1 + n_2}$

• Test Statistic (Pooled): $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\bar{p}(1-\bar{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

• Assumptions: Independent samples, SRS, sample size conditions.

• Example: Comparing proportions of success in two groups.

One-Sample Mean (Known and Unknown SD)

• Known SD: $z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$

• Unknown SD: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ with df = n-1

• Assumptions: Independent observations, SRS, nearly normal distribution.

• Example: Testing if the average weight differs from a known value.

Two-Sample Mean Tests (Equal and Unequal Variance)

• Unequal Variance: $t = \frac{\bar{x}_1 - \bar{x}_2 - \mu_0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

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

• Equal Variance: $t = \frac{\bar{x}_1 - \bar{x}_2 - \mu_0}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

• Pooled Variance: $s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}$

• Assumptions: Independent samples, SRS, nearly normal distribution.

• Example: Comparing mean scores between two classes.

Paired Sample t-Test

• Estimate: $\bar{d}$ (mean of differences)

• Standard Error: $\frac{s_d}{\sqrt{n}}$

• Test Statistic: $t = \frac{\bar{d} - \mu_0}{s_d/\sqrt{n}}$

• Assumptions: Differences have nearly normal distribution.

• Example: Before-and-after measurements on the same subjects.

Testing Categorical Data (Chi-Square Tests)

Overview

Chi-square tests are used to analyze categorical data, testing hypotheses about distributions or relationships between variables.

• Goodness of Fit: Tests if observed frequencies match expected frequencies.

• Independence/Homogeneity: Tests if two categorical variables are independent or if distributions are the same across groups.

Chi-Square Goodness of Fit

• Degrees of Freedom: # cells - 1

• Expected Frequency: $Exp_i = n \times p_i$

• Test Statistic: $\chi^2_{stat} = \sum \frac{(Obs - Exp)^2}{Exp}$

• Null Hypothesis: Population proportions are as specified.

• Assumptions: Count data, independence, SRS, large sample, expected counts ≥ 5.

Chi-Square Test of Independence/Homogeneity

• Degrees of Freedom: (r-1)(c-1)

• Expected Frequency: $Exp_{ij} = \frac{Row\ total_i \times Column\ total_j}{Table\ total}$

• Test Statistic: $\chi^2_{stat} = \sum \frac{(Obs - Exp)^2}{Exp}$

• Null Hypothesis: Variables are independent.

Testing for Non-Normal Distributions (Nonparametric Tests)

Overview

Nonparametric tests are used when data do not meet the assumptions of normality. They are based on ranks rather than actual values.

• Wilcoxon Signed-Rank Test: Tests median against a hypothesized value.

• Paired Wilcoxon Signed-Rank: Tests median of differences.

• Mann-Whitney U (Wilcoxon Sum Rank): Compares distributions of two independent samples.

Wilcoxon Signed-Rank Test

• Null Hypothesis: Median = θ0

• Test Statistic: $T = \min(T^-, T^+)$ where $T^-$ and $T^+$ are sums of ranks of negative and positive differences.

• Assumptions: Independent observations, SRS, reasonably symmetric distribution.

Mann-Whitney U Test

• Null Hypothesis: Two populations have identical distributions.

• Test Statistic: $W = T(\bar{n})$ where $\bar{n} = \min(n_1, n_2)$

• Assumptions: Independent observations, SRS, reasonably symmetric distributions.

Hypothesis Testing Decision Rules

Overview

Decision rules specify how to interpret test statistics and p-values to accept or reject hypotheses.

• One-sided tests: For Ha: parameter > value, reject if test statistic > critical value.

• Two-sided tests: For Ha: parameter ≠ value, reject if |test statistic| > critical value.

• Chi-square tests: Reject if $\chi^2_{stat} > \chi^2_{\alpha, df}$

• Nonparametric tests: Reject if test statistic falls in critical region.

Confidence Interval Formulas

Overview

Confidence intervals provide a range of plausible values for population parameters based on sample statistics.

• Two-sided CI: $Estimate \pm z_{\alpha/2} \times SD$ or $Estimate \pm t_{\alpha/2, df} \times SE$

• One-sided upper bound CI for means: $(-\infty, Estimate + z_{\alpha} \times SD)$ or $(-\infty, Estimate + t_{\alpha, df} \times SE)$

• One-sided upper bound CI for proportions: $(0, Estimate + z_{\alpha} \times SD)$

• One-sided lower bound CI for means: $(Estimate - z_{\alpha} \times SD, \infty)$ or $(Estimate - t_{\alpha, df} \times SE, \infty)$

• One-sided lower bound CI for proportions: $(Estimate - z_{\alpha} \times SD, 1)$

Margin of Error and Sample Size Calculations

Overview

Margin of error quantifies the uncertainty in an estimate. Sample size formulas help determine how large a sample is needed to achieve a desired margin of error.

• Margin of Error (ME): The maximum expected difference between the estimate and the true parameter.

• Sample Size for Proportions: $n = \left( \frac{z_{\alpha/2} \sqrt{\hat{p}(1-\hat{p})}}{ME} \right)^2$

• Sample Size for Means: $n = \left( \frac{z_{\alpha/2} \sigma}{ME} \right)^2$

Summary Table: Hypothesis Test Procedures

Test	Estimate	Standard Error	Null Hypothesis	Test Statistic	Assumptions
1-proportion z-test	\( \hat{p} \)	$\sqrt{\frac{\hat{p}\hat{q}}{n}}$	\( p = p_0 \)	$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$	Independence, SRS, sample size
2-proportion z-test (unpooled)	\( \hat{p}_1 - \hat{p}_2 \)	$\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$	\( p_1 - p_2 = p_0 \)	$z = \frac{\hat{p}_1 - \hat{p}_2 - p_0}{\text{SE}}$	Independence, SRS, sample size
2-proportion z-test (pooled)	\( \hat{p}_1 - \hat{p}_2 \)	$\sqrt{\bar{p}(1-\bar{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$	\( p_1 - p_2 = 0 \)	$z = \frac{\hat{p}_1 - \hat{p}_2}{\text{SE}}$	Independence, SRS, sample size
1-sample mean (known SD)	\( \bar{x} \)	$\frac{\sigma}{\sqrt{n}}$	\( \mu = \mu_0 \)	$z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$	Independence, SRS, normality
1-sample mean (unknown SD)	\( \bar{x} \)	$\frac{s}{\sqrt{n}}$	\( \mu = \mu_0 \)	$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$	Independence, SRS, normality
2-sample mean (unequal variance)	\( \bar{x}_1 - \bar{x}_2 \)	$\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$	\( \mu_1 - \mu_2 = \mu_0 \)	$t = \frac{\bar{x}_1 - \bar{x}_2 - \mu_0}{\text{SE}}$	Independence, SRS, normality
2-sample mean (equal variance)	\( \bar{x}_1 - \bar{x}_2 \)	$s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$	\( \mu_1 - \mu_2 = \mu_0 \)	$t = \frac{\bar{x}_1 - \bar{x}_2 - \mu_0}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$	Independence, SRS, normality
Paired t-test	\( \bar{d} \)	$\frac{s_d}{\sqrt{n}}$	\( \mu_d = \mu_0 \)	$t = \frac{\bar{d} - \mu_0}{s_d/\sqrt{n}}$	Independence, SRS, normality of differences

Summary Table: Chi-Square and Nonparametric Tests

Test	Null Hypothesis	Test Statistic	Assumptions
Chi-square Goodness of Fit	Population proportions as specified	$\chi^2_{stat} = \sum \frac{(Obs - Exp)^2}{Exp}$	Count data, independence, SRS, expected counts ≥ 5
Chi-square Independence/Homogeneity	Variables are independent	$\chi^2_{stat} = \sum \frac{(Obs - Exp)^2}{Exp}$	Count data, independence, SRS, expected counts ≥ 5
Wilcoxon Signed-Rank	Median = θ0	$T = \min(T^-, T^+)$	Independence, SRS, symmetric distribution
Mann-Whitney U	Identical distributions	$W = T(\bar{n})$	Independence, SRS, symmetric distributions

Additional info:

• "SRS" stands for Simple Random Sample.

• "n < 0.1N" means the sample size is less than 10% of the population, ensuring independence.

• Critical values (zα, tα,df, χ2α,df) are determined by the significance level (α) and degrees of freedom.

• For hypothesis tests, p-values are calculated based on the test statistic and compared to α to make decisions.