Comprehensive Guide to Parametric Statistics and the Student's t-test

Purpose of Inferential Statistics: The central aim is to generalize findings derived from a specific sample to a broader population.
Classification of Statistics: Inferential methods are categorized into two primary types:
- Parametric Statistics: Tests that make specific assumptions about the parameters of the population distribution.
- Non-parametric Statistics: Equivalent tests used when parametric assumptions are not met.
Advantages of Parametric Tests:
- They typically possess more statistical power than their non-parametric equivalents, meaning they are more likely to detect significant differences when they truly exist.
- Every parametric test has a corresponding non-parametric alternative.
Core Assumption: The most frequent assumption is that the population data from which the sample is drawn follow a normal distribution.

Definition: The "Student's t-test" (or simply the t-test) is a fundamental parametric test used in statistical analysis.
General Assumptions for Application:
- Quantitative Data: The dependent variable must be continuous (numerical: interval or ratio data).
- Random Selection: The sample must be selected randomly.
- Normality: The data must follow a normal distribution curve.
- Homogeneity of Variance: Samples should have equal variance.
Required Variables:
- Dependent Variable: Must be at the interval or ratio level.
- Independent Variable: Must be a nominal variable with exactly two groups.

Independent Sample t-test: Also known as the unpaired t-test or two-sample t-test.
Paired Sample t-test: Also referred to as the repeated measures test, dependent samples t-test, matched pairs t-test, or paired-difference t-test.
One Sample t-test: Used to compare a sample against a standard or hypothesized value.

Purpose: To compare the means of two unrelated (independent) groups to determine if there is a statistically significant difference between their population means.
Independence Property: The data from one group does not affect the data from the other group. The two samples consist of different test subjects.
Application Criteria:
- Comparing the average (mean) of a continuous variable between two distinct groups.
- Examples include comparing males versus females or a treatment group versus a placebo group.
Hypothetical Scenarios:
- Comparing effectiveness between two different teaching methods based on exam scores.
- Investigating salary differences between men and women within the same profession.
- Testing blood pressure differences between a medication group and a control group.
- Assessing the average height of men against the average height of women.
Energy Drink Example (Unpaired Design):
- Measuring the heart rate of $10$ people before they consume an energy drink and then measuring the heart rate of a completely different group of people who have already consumed energy drinks.

Definition: A test designed to compare the means of the same group or item under two separate scenarios.
Design Characteristics:
- Involves individual observations from the first sample matched to observations in the second sample (often the same person).
- Typical design: Before-and-after measurements for each subject.
- Interventions are performed at two different points in time on the same sample.
Key Statistical Features:
- Variance: Variance is not assumed to be equal because the sample is the same group measured twice.
- Sample Size: It is performed on two random samples of the same size.
- Equivalence: A paired t-test is statistically equivalent to a one-sample t-test performed on the differences.
- Power: This test is more powerful than the independent t-test because consistency in test subjects provides more information about the relationship between variables.
Energy Drink Example (Paired Design):
- Measuring the heart rate of $10$ specific individuals before they drink an energy drink and then measuring the heart rate of those same $10$ individuals after they drink the energy drink.

Simple random sampling must be utilized.
Sample sizes are frequently small.
Two measurements are drawn from the same pair of objects or individuals.
The test is performed on the "differences" calculated from the matched pairs.
Normality Requirement: Either the population of differences must be normally distributed, or the number of differences must be large enough ( $n$ is sufficiently large) so the distribution of the sample mean of differences is approximately normal.

Purpose: To determine if a single sample's mean is statistically different from a known population mean or a specific hypothesized value.
Usage Context: Use this test when you want to know if a sample comes from a particular population but full population data is unavailable, or when comparing an average to a fixed external target (e.g., national averages).
Examples:
- IQ Comparison: If the average IQ of college students is $100$ and a sample of dental students has an average IQ of $128$ , the one-sample t-test determines if dental students are significantly smarter than the general college population.
- Acidity Testing: Comparing the average pH of a liquid sample to the standard neutral pH of $7$ .

Null Hypothesis ( $H_0$ ): The assumption that there is no difference between the means of two population groups. The expected difference in population mean is zero.
Alternative Hypothesis ( $H_1$ ): The assertion that a statistically significant difference exists between the means.
Example Research Question: Is there a difference in birth weights of children born to $15$ non-smokers compared to those born to $14$ heavy smokers?
- $H_0$ : There is no difference in birth weight between the two groups.
- $H_1$ : A difference in birth weight between the two groups exists.

Sample Data Breakdown:
- Non-smokers group: $15$ subjects; Mean birth weight = $3.59\,kg$ .
- Heavy smokers group: $14$ subjects; Mean birth weight = $3.2\,kg$ .
- Observation: The standard deviation (SD) is smaller for the non-smoking group compared to the heavy smoking group.
Box Plot Interpretation: A box plot provides the shape of the distribution, identifying:
- Minimum Value and Maximum Value.
- Median (Middle line).
- Quartiles: $Q1$ ( $25\%$ of observations), $Q2$ (Median), and $Q3$ ( $75\%$ of observations).
- Inter-quartile Range (IQR): The distance between $Q1$ and $Q3$ .
Levene’s Test for Homogeneity of Variance:
- This test is included in SPSS output to verify the assumption of equal variance.
- Rule 1: If the F-value is significant ( $p < 0.05$ ), variances are different (assumption violated). Use the t-statistic on the bottom line of the output table.
- Rule 2: If the F-value is not significant ( $p > 0.05$ ), variances are equal. Use the t-statistic on the top line of the output table.
Results and Conclusion:
- Resulting p-value: $0.022$ .
- Significance: Since $p = 0.022$ ( $p < 0.05$ ), the difference between means is significant.
- Conclusion: Children born to non-smokers are, on average, heavier than those born to heavy smokers by approximately $0.39\,kg$ (noted as $0.39\,gr$ in context of difference).