Basic Statistics - Hypothesis Testing

ON TEENAGERS, ADULY

  • Statistics indicate that teenage pregnancy rates drop significantly after age 25.

  • Cited contributor: Mary Anne Tebeds, Regalicon state senator from Cetorado Springs, commented by Harry F. Punce.

BASIC STATISTICS FOR THE BEHAVIORAL SCIENCES

  • Author: Gary W. Heiman

  • 5th edition

  • Chapter Ten: Introduction to Hypothesis Testing

NEW STATISTICAL NOTATION

  • Greater than: >

  • Less than: <

  • Greater than or equal to: ≥

  • Less than or equal to: ≤

  • Not equal to: ≠

THE ROLE OF INFERENTIAL STATISTICS IN RESEARCH

  • Inferential statistics are pivotal in determining whether observed relationships in samples reflect true relationships within the larger population.

RELATIONSHIPS IN EXPERIMENTS

  • As the conditions of the independent variable are altered (for example, varying drug doses such as 5mg, 10mg, 20mg), it's expected to observe a pattern of differences in the dependent variable (for example, measuring pain relief).

SAMPLING ERROR

  • Definition: Sampling error occurs when random chance results in a statistic from the sample that does not accurately represent the population parameter.

  • Clarification: A sample's mean, median, mode, standard deviation may be so unusual that it clearly does not represent the population from which it was drawn.

RELATIONSHIPS & SAMPLING ERROR

  • Example Scenario 1:

    • If an experiment is conducted on gender and artistic ability, and the sample inadvertently contains many highly artistic men and less artistic women, the conclusion drawn could wrongfully indicate a difference in creativity between genders.

  • Example Scenario 2:

    • Conducting research on gender and height might yield misleading results if the sample includes unusually tall women and short men, possibly suggesting incorrect conclusions about average height differences between genders.

  • Implication: Sampling error can lead to misleading interpretations, including falsely identifying relationships where none exist or vice versa.

INFERENTIAL STATISTICS

  • Purpose: Inferential statistics help assess whether a relationship observed in a sample is representative of the overall population or if it's a product of random sampling error.

PARAMETRIC STATISTICS

  • Definition: Statistics that assume certain parameters about the population being represented must be met.

  • Common parameters for parametric tests:

    • The distribution of dependent scores should form a normal distribution.

    • Scores must be measured on an interval or ratio scale.

NONPARAMETRIC PROCEDURES

  • Definition: Nonparametric statistics are inferential procedures that do not require strict assumptions about the populations being represented.

  • Applicability: These procedures are useful with nominal or ordinal data, or skewed interval or ratio data.

ROBUST PROCEDURES

  • Explanation: Parametric procedures are termed robust because even if the data doesn't meet the assumptions perfectly, it results in only a negligible amount of error in inferences.

SETTING UP INFERENTIAL PROCEDURES

  • Steps in setting up an experiment:

    1. Create a hypothesis.

    2. Design an experiment to test the hypothesis.

    3. Translate the hypothesis into a statistical hypothesis.

    4. Select the appropriate statistical procedure to test the hypothesis.

EXPERIMENTAL HYPOTHESES

  • Definition: Experimental hypotheses predict outcomes of experiments.

  • Types:

    • Null hypothesis: posits no relationship (the default).

    • Alternative hypothesis: posits that a relationship does exist.

PREDICTING A RELATIONSHIP

  • Two-tailed test: Used when predicting a relationship without knowing the direction (e.g., increase or decrease).

  • One-tailed test: Used when predicting the direction of a relationship.

A ONE-SAMPLE EXPERIMENT

  • In performing a one-sample experiment, it is required to have prior knowledge of the population mean under different conditions of the independent variable.

  • Example: Administering a “smart pill” to 100 people and measuring resulting IQs where the known population mean is established (e.g., typically, a population mean IQ might be 100).

CREATE YOUR STATISTICAL HYPOTHESIS

  • To validate the experiment, translate the theory into a statistical hypothesis, evaluating potential success and failure in numeric terms.

NULL HYPOTHESIS

  • Representation: H_0: ext{m} = 100

  • Meaning: Predicts no effect of the drug on IQ, thus suggesting the sample mean will be the same as the population average (100).

ALTERNATIVE HYPOTHESIS

  • Representation: H_a: ext{m}
    eq 100

  • Meaning: Predicts the sample mean will differ from the population average (100).

LOGIC OF STATISTICAL TESTING

  • Hypothetical scenario: If a sample's mean IQ is 105, one might wrongly attribute this as evidence the pill works, while in reality, it could be a result of drawing an atypical sample due to sampling error.

PERFORMING THE z-TEST

  • Definition: The z-test computes a z-score for a sample mean to evaluate its position on a sampling distribution of means.

ASSUMPTIONS OF THE z-TEST

  • Conditions include:

    1. Randomly selected sample.

    2. Dependent variable should be at least approximately normally distributed.

    3. Knowledge of the population mean under a condition of the independent variable.

    4. True standard deviation of the population must be known.

SETTING UP FOR A TWO-TAILED TEST

  • Process involves determining the alpha level (e.g., typically ext{p} = 0.05), identifying the rejection region, and determining the critical value (critical z-value).

  • Critical z-value for two-tailed tests: ext{z}_{crit} = ext{±}1.96.

REJECTING H0

  • If z_{obt} lies in the rejection region (exceeds the critical value), hypothesis {H_0} is rejected in favor of the alternative hypothesis {H_a}, indicating significant results.

  • Note: Significant results indicate that the outcomes are unlikely to have occurred due to chance, not necessarily that they are important or useful.

INTERPRETING SIGNIFICANT RESULTS

  • Rejecting H_0 does not equate to proving H_0 false; it merely indicates results unlikely to occur by chance.

FAILING TO REJECT H0

  • If z_{obt} does not fall in the region of rejection, we do not reject H_0.

  • These results are termed non-significant, suggesting any observed differences could result from sampling error, implying no true relationship exists within the population.

INTERPRETING NONSIGNIFICANT RESULTS

  • Failing to reject H_0 does not prove it true, similar to a jury declaring a defendant not guilty without proving innocence.

SUMMARY OF THE z-TEST

  1. Determine experimental hypotheses (theory) and create statistical hypothesis (math).

  2. Compute z_{obt}.

  3. Set up sampling distribution for H_{0} (identify rejection regions and critical values).

  4. Compare z_{obt} with critical z-value (z_{crit}).

THE ONE-TAILED TEST

  • Definition: Utilized when a predicted direction of the effect on scores is established.

ONE-TAILED HYPOTHESES

  • When an increase is predicted:

    • Null hypothesis: H_0: ext{m} ext{≤} 100 (predicts no increase or potentially a decrease).

    • Alternative hypothesis: H_a: ext{m} > 100 (predicts an increase).

  • When a decrease is predicted:

    • Null hypothesis: H_0: ext{m} ext{≥} 100 (predicts no decrease or potential increase).

    • Alternative hypothesis: H_a: ext{m} < 100 (predicts a decrease).

PRACTICE PROBLEMS

  • Problem 1: Determine hypotheses for IQ-related television watching (2.5 hours). ( H_0: m ) and ( H_a: m ).

  • Problem 2: Assess Pepperdine students in memory tests (average of 7 numbers). ( H_0: m ) … and ( H_a: m ) ….

ERRORS IN STATISTICAL TESTING

TYPE I ERRORS

  • Definition: Falsely rejecting H_0 when it is true.

  • Result: An unlikely sample prompts incorrect belief in a relationship that does not exist.

  • Probability: Theoretical probability of the type I error affirms as ( ext{α} ).

TYPE II ERRORS

  • Definition: Retaining H_0 when it is false.

  • Result: Sample produces typical scores suggesting a non-existent relationship actually does exist.

  • Probability: The theoretical probability of a type II error is represented as ( ext{β} ).

SUMMARY OF ERRORS

  • Error types are depicted in a comparative table showing false positives (Type I) and false negatives (Type II).

BALANCING ACT IN ERRORS

  • Key takeaway: Minimizing Type I errors often leads to under-powering the study, increasing the risk of Type II errors.

  • Researcher strategies include:

    • Utilizing lower alpha levels to minimize Type I errors.

    • Structuring studies to maximize statistical power.

POWER IN RESEARCH

  • Definition: Power refers to the probability of correctly rejecting H_0 when it is in fact false.

  • Strategies to increase power:

    • Design studies using parametric procedures.

    • Utilize directionally predictive one-tailed tests.

Zcrits IN ONE VS. TWO TAILED TESTS

  • In one-tailed tests, the critical z-value is often closer to the mean, allowing samples to be less extreme to qualify for rejection of H_0.