Research Methods 4

Null Hypothesis ($H_0$)

Predicts that no difference exists between the groups or conditions being compared.

Alternative Hypothesis ($H_1$)

Predicts that a difference exists between the groups or conditions being compared.

Rejecting the Null Hypothesis

When experimental results are statistically significant, $H_0$ is rejected, providing empirical support for $H_1$.

Failing to Reject the Null Hypothesis

When results are not statistically significant, there is insufficient evidence to reject $H_0$, though this does not prove $H_0$ is true.

Type I Error

Rejecting the null hypothesis when it is actually true; also known as a false alarm.

Type II Error

Failing to reject the null hypothesis when it is actually false; also known as a miss.

Statistical Power

The probability of correctly rejecting the null hypothesis when it is false and a real effect exists.

Alpha ($\alpha$)

The significance level, representing the chosen probability of committing a Type I error during testing.

Typical Alpha Level in Psychology

$\alpha = .05$, representing a $5\%$ risk of committing a Type I error.

Directional Hypothesis

An alternative hypothesis that predicts both a difference and the specific direction of that difference.

Nondirectional Hypothesis

An alternative hypothesis predicting a difference between groups without specifying which group will score higher or lower.

One-Tailed Test

A test used for a directional hypothesis where the rejection region is located entirely within one tail of the distribution.

Two-Tailed Test

A test used for a nondirectional hypothesis where the rejection region is split equally between both tails of the distribution.

Rejection Region

The area of the sampling distribution containing values sufficiently extreme to reject $H_0$ at the chosen alpha level.

Rejection Region in a One-Tailed Test ($\alpha = .05$)

Contains $5\%$ of the total distribution area positioned entirely in one tail.

Rejection Region in a Two-Tailed Test ($\alpha = .05$)

Contains $2.5\%$ of the distribution area in each tail, totaling $5\%$.

Positive Directional Hypothesis

Predicts that the sample mean will exceed the population mean, placing the rejection region in the right tail.

Negative Directional Hypothesis

Predicts that the sample mean will be less than the population mean, placing the rejection region in the left tail.

Z-Test

A statistical test used to determine whether a sample mean significantly differs from a known population mean.

Requirements for a Z-Test

The population standard deviation must be known and the sample size must be large ($N \ge 30$).

Z-Statistic

Converts an observed sample mean into a standardized score on the standard normal distribution.

Z-Test Formula

$$z = \frac{\bar{X} - \mu}{\sigma / \sqrt{N}}$$

Standard Error for a Z-Test

$SE = \frac{\sigma}{\sqrt{N}}$, calculated using the known population standard deviation.

Population Mean ($\mu$)

The average score of the entire target population used as the baseline for comparison.

Sample Mean ($\bar{X}$)

The average score computed from the individuals measured in the sample.

Population Standard Deviation ($\sigma$)

A parameter quantifying variability in the population, used to calculate the standard error for a $z$-test.

Sample Size ($N$)

The total number of observations, scores, or individual participants in the sample.

Central Limit Theorem

As sample size increases, the sampling distribution of the mean approaches normality, regardless of the population shape.

Critical Z-Value for a Two-Tailed Test ($\alpha = .05$)

$$\pm 1.96$$

Critical Z-Value for a Positive One-Tailed Test ($\alpha = .05$)

$$+1.645$$

Critical Z-Value for a Negative One-Tailed Test ($\alpha = .05$)

$$-1.645$$

Significant Z-Statistic

A calculated value falling within the rejection region, providing sufficient evidence to reject $H_0$.

Nonsignificant Z-Statistic

A calculated value falling outside the rejection region, resulting in a failure to reject $H_0$.

Z-Table

A reference table used to determine statistical significance by pairing a calculated $z$-value with critical cutoffs or proportions.

Positive Z-Statistic

Indicates that the observed sample mean is greater than the hypothesized population mean.

Negative Z-Statistic

Indicates that the observed sample mean is less than the hypothesized population mean.

APA Reporting for a Significant Z-Test

Report the sample and population means, the test statistic, and the probability value (e.g., $z = 4.00$, $p < .05$).

APA Reporting for a Nonsignificant Z-Test

Report the means, the test statistic, and $p > .05$, documenting the lack of statistical significance.

One-Sample T-Test

A test assessing whether a sample mean differs from a population mean when the population standard deviation is unknown and estimated from the sample.

When to Use a T-Test

When the population standard deviation is unknown and must be estimated via the sample standard deviation, particularly with small samples.

T-Test Formula

$$t = \frac{\bar{X} - \mu}{s / \sqrt{N}}$$

Standard Error for a T-Test

$SE = \frac{s}{\sqrt{N}}$, calculated using the sample standard deviation.

Difference Between Z-Test and T-Test

The $z$-test requires the population standard deviation ($\sigma$), whereas the $t$-test utilizes the sample standard deviation ($s$).

T-Distribution

A family of symmetrical, bell-shaped distributions whose shapes vary according to degrees of freedom.

Why the T-Distribution Is a Family of Distributions

Varying degrees of freedom produce curves with different peak heights and tail thicknesses.

Small Degrees of Freedom in T-Distributions

Produces a shorter central peak with thicker, heavier tails relative to the standard normal curve.

Large Degrees of Freedom in T-Distributions

Produces a curve that increasingly approximates the standard normal distribution.

Degrees of Freedom ($df$)

The number of values in a final calculation of a statistic that are free to vary.

Degrees of Freedom for a One-Sample T-Test

$$df = N - 1$$

T-Table

A table used to locate critical values based on degrees of freedom, alpha level, and directional nature of the test.

Critical T-Value

The threshold value that an empirical $t$-statistic must meet or exceed in the predicted direction to reject $H_0$.

Significant T-Statistic

A calculated value falling within the rejection region, providing justification to reject $H_0$.

Nonsignificant T-Statistic

A calculated value falling short of the rejection region, resulting in failure to reject $H_0$.

Directional T-Test

A one-tailed test in which the sign of the calculated $t$-statistic must match the predicted direction of the effect.

Positive Directional T-Test

Predicts that the sample mean exceeds the population mean, requiring a sufficiently positive $t$-statistic to reject $H_0$.

Negative Directional T-Test

Predicts that the sample mean is below the population mean, requiring a sufficiently negative $t$-statistic to reject $H_0$.

Degrees of Freedom Not Listed in T-Table

Use the critical value for the next lower degrees of freedom listed in the table to maintain conservative testing.

APA Reporting for a Significant T-Test

Report the sample mean, standard deviation, degrees of freedom, $t$-statistic, and significance level (e.g., $t(29) = 2.34$, $p < .05$).

APA Reporting for a Nonsignificant T-Test

Report the degrees of freedom, $t$-statistic, and nonsignificance notation (e.g., $t(7) = 0.565$, n.s.).

Statistical Power (Detection View)

The probability of correctly detecting a real effect by rejecting a false null hypothesis.

Four Ways to Increase Statistical Power

Employ a justified one-tailed test, increase sample size, strengthen dose or exposure, and decrease score variability.

One-Tailed Tests and Power

Increases power because the critical cutoff is placed closer to the distribution center in the predicted direction.

Sample Size and Power

Increasing sample size decreases standard error, enhancing the sensitivity to detect genuine treatment effects.

Dose or Exposure and Power

Strengthening the independent variable elevates effect size, making true group differences easier to detect.

Variability and Power

Reducing extraneous variation and measurement noise sharpens detection of real statistical effects.

Standard Error and Sample Size Relationship

As sample size increases, standard error decreases inversely with the square root of $N$.

Effect Size

A quantitative metric of the magnitude of a difference or relationship, independent of sample size.

Correlation Coefficient ($r$)

A standardized index measuring the strength and direction of the linear relationship between two continuous variables.

Positive Correlation

A relationship pattern where both variables tend to increase or decrease together in the same direction.

Negative Correlation

An inverse relationship pattern where increases in one variable correspond with decreases in the other.

Correlation Strength

Determined strictly by the absolute value $|r|$, regardless of whether the sign is positive or negative.

Weak Correlation Range

An absolute $r$-value ranging from approximately $.00$ to $.29$.

Moderate Correlation Range

An absolute $r$-value ranging from approximately $.30$ to $.69$.

Strong Correlation Range

An absolute $r$-value ranging from approximately $.70$ to $1.00$.

Degrees of Freedom for Correlation

$$df = N_{\text{pairs}} - 2$$

R-Table

A table used to assess whether an observed correlation coefficient reaches statistical significance given the degrees of freedom.

Significant Correlation

An observed correlation whose absolute magnitude reaches or exceeds the critical value at the specified alpha level.

Nonsignificant Correlation

An observed correlation falling below the critical threshold, indicating insufficient evidence of a population relationship.

APA Reporting for Correlation

Report the degrees of freedom, correlation coefficient, and significance level (e.g., $r(18) = -.56$, $p < .05$).

Correlation and Causation

A statistically significant correlation indicates an association but cannot demonstrate that one variable causes changes in the other.

Third-Variable Problem

An unmeasured third factor that influences both variables, generating or explaining their observed correlation.

Directionality Problem

The ambiguity in determining which correlated variable is influencing or causing changes in the other.

Hewitt and Homan (2004) Focus

Examined whether question readability difficulty in standardized tests was related to third-grade student performance.

Hewitt and Homan Correlation Finding

Found a moderate negative correlation ($r = -.56$) between question readability difficulty and the proportion of correct student answers.

Hewitt and Homan Interpretation

Higher reading difficulty corresponded to fewer correct responses, but the correlational design did not establish causal direction.

Between-Subjects Experiment

An experimental design where different participants are assigned to separate groups representing distinct levels of the independent variable.

Within-Subjects Experiment

An experimental design where each individual participant is exposed to every level of the independent variable.

Independent Variable in Hewitt and Homan's Proposed Experiment

Question wording format: original test questions versus simplified, easier-to-read versions.

Dependent Variable in Hewitt and Homan's Proposed Experiment

Student test performance scores on the Social Studies subscale.

Nuremberg Code

An early international research ethics code; requires identifying its establishment year and five core principles relevant to psychological research.

Declaration of Helsinki

An international ethical policy detailing required safeguards for research involving human subjects who are legally incompetent.

Belmont Report

A seminal ethical framework establishing three fundamental principles (respect for persons, beneficence, and justice) for human subjects research.