7.4 Single-Sample Tests for Means, Proportions, and Variances

Single-sample tests are crucial tools in statistical inference. They help us compare a sample statistic to a hypothesized population parameter, allowing us to make informed decisions about population characteristics based on sample data.

These tests include t-tests for means, z-tests for proportions, and chi-square tests for variances. Each test has specific assumptions and procedures, guiding researchers in selecting the most appropriate method for their data and research questions.

Single-Sample Tests for Means

T-test for population mean

• Compares sample mean to hypothesized population mean tests null hypothesis about population mean
• Assumes normally distributed population or large sample size random sampling continuous or interval-level data
• Test statistic calculated using $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ where $\bar{x}$ is sample mean $\mu_0$ is hypothesized population mean $s$ is sample standard deviation $n$ is sample size
• Degrees of freedom $df = n - 1$ impact critical values and p-values
• Hypothesis testing procedure:

1. State null and alternative hypotheses
2. Choose significance level
3. Calculate test statistic
4. Determine p-value or compare to critical value
5. Make decision to reject or fail to reject null hypothesis

• Interpret results considering statistical significance practical significance confidence intervals for mean difference

Z-test for population proportion

• Compares sample proportion to hypothesized population proportion tests null hypothesis about population proportion
• Assumes large sample size (np and n(1-p) both > 5) random sampling binary or categorical data
• Test statistic calculated using $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ where $\hat{p}$ is sample proportion $p_0$ is hypothesized population proportion $n$ is sample size
• Uses standard normal distribution for determining p-values and critical values
• Follows same hypothesis testing procedure as t-test
• Interpret results considering statistical significance practical significance confidence intervals for proportion difference

Single-Sample Tests for Variances

Chi-square test for population variance

• Compares sample variance to hypothesized population variance tests null hypothesis about population variance
• Assumes normally distributed population random sampling continuous data
• Test statistic calculated using $\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$ where $n$ is sample size $s^2$ is sample variance $\sigma_0^2$ is hypothesized population variance
• Degrees of freedom $df = n - 1$ impact critical values and p-values
• Chi-square distribution right-skewed approaches normal distribution as df increases
• Follows same hypothesis testing procedure as t-test and z-test
• Interpret results considering statistical significance practical significance confidence intervals for variance ratio

Selection of single-sample tests

• Consider type of data (continuous, categorical, binary) sample size population distribution parameter of interest (mean, proportion, variance)
• Use decision tree: continuous data (t-test for mean, chi-square test for variance) categorical data (z-test for proportion)
• Sample size considerations: large samples (z-test for means) small samples (t-test for means)
• T-test robust to violations of normality z-test sensitive to sample size assumptions
• Meeting test assumptions impacts Type I and Type II errors validity of results and conclusions
• Consider alternative tests for non-normal data (Wilcoxon signed-rank test, bootstrap methods)