T_tests_and_Z_tests.md
T-tests and Z-tests are inferential statistical tests used to compare means. They assess whether observed differences between groups are statistically significant.
1. When to Use
Comparing mean blood pressure between treatment and control groups
Evaluating changes in performance before and after intervention
Determining if a sample mean differs from a known population mean
2. T-tests
Used when the population standard deviation is unknown and sample size is small.
a. One-sample t-test
Compares the sample mean to a known or hypothesized population mean.
# One-sample t-test
t.test(sample_data, mu = 100)b. Independent two-sample t-test
Compares means between two independent groups.
# Two-sample t-test
t.test(group1, group2)c. Paired t-test
Used for paired or matched samples.
# Paired t-test
t.test(before, after, paired = TRUE)Assumptions
Data are continuous and approximately normally distributed
Homogeneity of variances (for independent samples)
Observations are independent
3. Z-tests
Used when population variance is known or sample size is large (n > 30).
a. One-sample Z-test
Compares a sample mean to a known population mean with known population standard deviation.
b. Two-sample Z-test
Compares means from two independent large samples with known variances.
Z-tests are less commonly used in practice because population standard deviation is rarely known.
4. Interpreting Results
Null hypothesis (H₀): The means are equal
Alternative hypothesis (H₁): The means are different
If p-value < 0.05, reject H₀ and conclude a significant difference exists
5. Summary
T-tests and Z-tests are fundamental tools in hypothesis testing for means. T-tests are preferred when standard deviation is unknown. Choose the test based on sample design and variance knowledge.
Last updated