Hypothesis Testing: Z-tests and T-tests

• This presentation discusses hypothesis testing, focusing on z-tests and t-tests.
• It is presented by Sir Reneil.

Hypothesis Testing

• A hypothesis is a statement about a population parameter subject to verification.
• Hypothesis testing assesses the validity of a hypothesis.
• A null hypothesis (H0) is a statement of no effect or no difference, it is the hypothesis to be tested.
• An alternative hypothesis (H1) represents the opposite of the null hypothesis, what the researcher wants to prove.
• A test statistic is a value calculated from sample data used to determine whether to reject the null hypothesis.
• The rejection region is the set of values for the test statistic that leads to rejection of the null hypothesis.
• A type I error occurs when the null hypothesis is rejected when it is true, also known as a false positive.
• The probability of a type I error is denoted by α (alpha), also called the significance level.
• A type II error occurs when the null hypothesis is not rejected when it is false, also known as a false negative.
• The probability of a type II error is denoted by β (beta).
• Statistical significance means the observed result is unlikely to have occurred by chance alone if the null hypothesis is true.
• A critical value is a point on the test distribution compared to the test statistic to determine whether to reject the null hypothesis.
• P-value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.

Z-Tests

• Z-tests are used to test hypotheses about a population mean when the population standard deviation is known and the sample size is large enough (n ≥ 30), or the population is normally distributed.
• The z-test statistic formula is: z = (x̄ - μ) / (σ / √n), where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

T-Tests

• T-tests are used to test hypotheses about a population mean when the population standard deviation is unknown and estimated from the sample.
• T-tests are appropriate for small sample sizes (typically n < 30).
• There are three types of t-tests: one-sample, independent samples, and paired samples.

One-Sample T-Test

• One-sample t-test compares the mean of a single sample to a known value or hypothesized population mean.
• The formula for the one-sample t-test statistic is: t = (x̄ - μ) / (s / √n), where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.
• Degrees of freedom (df) for a one-sample t-test is n - 1.

Independent Samples T-Test

• An independent samples t-test compares the means of two independent groups.
• The formula for the independent samples t-test statistic is more complex and involves the sample means, sample standard deviations, and sample sizes of both groups.
• Degrees of freedom (df) for an independent samples t-test is calculated based on the sample sizes of both groups.

Paired Samples T-Test

• A paired samples t-test compares the means of two related groups (e.g., before and after measurements on the same subjects).
• It is also known as a dependent samples t-test.
• The formula for the paired samples t-test statistic involves the mean difference between pairs of observations.
• Degrees of freedom (df) for a paired samples t-test is n - 1, where n is the number of pairs.

This presentation explains hypothesis testing, covering null and alternative hypotheses, test statistics, and rejection regions. It also addresses Type I and Type II errors. Focus is given to z-tests and t-tests.