Introduction to Statistical Inference and Hypothesis Testing

Introduction to Statistical Inference and Hypothesis Testing

  • Overview of Topics

    • Descriptive and Inferential statistics

    • Sampling and Sampling error

    • The logic of hypothesis testing

    • Research questions and hypothesis construction

    • H0 (Null hypothesis) and H1 (Alternative hypothesis)

    • The hypothesis testing process

    • Alpha level and decision rules

Descriptive and Inferential Statistics

  • Descriptive Statistics

    • Definition: Procedures that summarize the pertinent characteristics of a set of data.

    • Examples of descriptive statistics include:

    • Mean

    • Median

    • Standard deviation (SD)

    • Interquartile range (IQR)

  • Inferential Statistics

    • Definition: Procedures that allow us to draw conclusions about the extent to which the characteristics of a sample data set are representative of a larger population.

Sampling and Sampling Error

  • Sampling

    • Definition: The process of drawing or selecting a sample from a population.

    • Importance: To apply inferential statistics, sampling must entail random selection.

    • Example:

    • Simple Random Sampling: Each member of the population has an equal chance of being selected for the sample.

  • Sampling Error

    • Definition: Occurs when a sample is not wholly representative of the population due to chance factors.

    • Example:

    • Consider a pot of beans (population) with various sizes. If a sample includes disproportionately large beans, the calculated sample mean may not represent the true mean of the whole population.

    • This deviation is referred to as Sampling Error.

Introduction to Hypothesis Testing

  • Hypothesis Testing

    • Definition: A systematic decision-making procedure to determine if the results of an experiment or survey support a specific theory or proposition.

  • Components of Hypothesis Testing

    • Empirical Methods:

    • Experiments

    • Observations (Naturalistic, Controlled, Participant)

    • Tests (Standardized/Objective, Projective)

    • Surveys and Questionnaires

    • Case Studies

  • The Research Process

    • Involves discovering general laws or principles.

The Hypothesis Testing Process

  • Steps in the Process

    1. Formulate the Research Hypothesis from theory, model, or observations.

    2. Design study to test the Research Hypothesis.

    3. Derive and state Null (H0) and Alternative Hypotheses (H1).

    4. Conduct study and test the Null Hypothesis.

    5. Reject (success) or fail to reject (inconclusive) the Null Hypothesis.

    6. Modify theoretical concepts based on the Alternative Hypothesis.

Types of Hypothesis

  • Research Hypothesis

    • Also known as the research question or research ‘idea’.

  • Null Hypothesis (H0)

    • Stated formally, represents no effect or no difference.

  • Alternative Hypothesis (H1)

    • Stated formally against the Null Hypothesis.

Stages of Hypothesis Testing

  1. Reframing the Research Question

    • Convert research question into H0 and H1, forming two opposing statements.

    • Example for correlation:

      • H0: There is no correlation between A and B

      • H1: There is a statistically significant correlation between A and B

    • Example for difference:

      • H0: There is no difference between A and B

      • H1: There is a statistically significant difference between A and B

  2. Assume Null Hypothesis is True

    • This is the basis for hypothesis testing.

    • Construct a distribution that reflects this assumption, often modeled as a normal distribution.

  3. Determine Cutoff Score

    • This represents the score at which the null hypothesis should be rejected.

  4. Determine Sample Score

    • Calculate the test statistic based on sample data and compare this score to the cutoff score.

  5. Decision Making

    • Compare scores from the previous two steps to decide on rejecting or not rejecting the null hypothesis.

Null and Alternative Hypotheses Examples

  • Example 1 (Drug Use and Memory)

    • Research Claim: Drug use impairs memory.

    • H0: There is no difference between drug users and non-drug users in memorizing and recalling words.

    • H1: There is a significant difference in memorization and recall abilities between these two groups.

  • Example 2 (SES and Test Scores)

    • Research Claim: Scores on the 11+ exam are correlated with socioeconomic status (SES).

    • H0: No correlation between 11+ scores and SES.

    • H1: There is a statistically significant correlation between these two variables.

Additional Samples and Analysis

  • Sample Hypothesis Development

    • a) Calculators Example:

    • Average mass of calculators is believed to be 450g.

    • Engineer measures average mass of 50 calculators and challenges this claim.

    • b) Graduation Rates:

    • Belief that at least 80% of students complete high school.

    • c) Average GPA Test:

    • Testing if students' average GPA differs from 2.7.

    • d) Vehicle Ownership Rates:

    • Researches a claim stating no more than 75% of residents own a vehicle.

Assumptions in Hypothesis Testing

  • The null hypothesis is always initially assumed to be true.

  • The resulting distribution from this assumption typically follows a normal shape.

  • Cutoff Scores

    • Commonly, a 0.05 (5%) significance level is utilized as the cutoff.

    • At times, 0.01 (1%) is used for more conservative significance levels, such as in drug trials.

Interpretation of Results

  • Results corresponding to significant different results lead to rejection of the null hypothesis.

  • If measured differences are attributed to sampling error, this indicates the results were due to chance without true differences.

Conclusion

  • A thorough understanding of hypothesis testing concepts is essential for statistical analysis.

  • The decision to reject or fail to reject the null hypothesis carries significant implications in research findings and applications.