PY2RMP 1) Review of Statistical Methods, Power, and Data Transformations

Comprehensive review flashcards covering research methodology, hypothesis testing, Type I and Type II errors, power analysis, test selection criteria, skewness, Simpson's paradox, ceiling/floor effects, and linear vs. non-linear transformations.

According to the research cycle model adapted from Hastie & Stasser (2000), which processes connect the "World" to "Theory" and "Data"?

The "World" connects to "Theory" via abstraction, and the "World" connects to "Data" via systematic observation & experimentation.

What is the distinction between categorical variables and continuous variables?

Categorical variables sort information into distinct groups (such as gender or marital status), whereas continuous variables are numeric measurements that can take on an infinite number of values (such as height or weight).

How can comfort level with military personnel be measured across categorical, ordinal, and interval data types?

Categorical: binary choice (e.g., Comfortable vs. Uncomfortable). Ordinal: ranked categories (e.g., Very comfortable, Somewhat comfortable, Neither comfortable nor uncomfortable, Somewhat uncomfortable, Very uncomfortable). Interval: numeric slider/scale (e.g., a score of $18.5$).

What is the Null Hypothesis ($H_0$) and what does it state?

The null hypothesis is assumed to be true in the population from which the sample is drawn; it states that there is no effect, no change, or no relationship between variables.

What does a significance level of $\alpha = 0.05$ mean in statistical testing?

It means the researcher accepts a $5\%$or lower probability of obtaining the test statistic (or one more extreme) in the sample by chance alone if the null hypothesis is true in the population.

What is the definition of a p-value?

The probability of finding the observed data due to chance alone if the null hypothesis is true.

What is a Type I error?

A Type I error (or false positive) occurs when the test result appears statistically significant (rejecting $H_0$) when there is actually no genuine effect in the population.

What is a Type II error?

A Type II error (or false negative) occurs when the test result appears nonsignificant (accepting/retaining $H_0$) when there is a genuine effect in the population.

What is the operational difference between a one-tailed test and a two-tailed test?

A one-tailed test evaluates an effect or relationship in a single specific direction, whereas a two-tailed test evaluates any change or difference in either direction.

What is statistical power, and how is it calculated?

Statistical power reflects whether a statistical test has enough capacity to detect an effect (given that all assumptions are met), calculated as $\text{Power} = 1 - \beta$. High power is typically between $0.8$ and $0.9$.

What is the relationship between effect size (Cohen's $d$) and the required sample size?

Effect size is directly proportional to power, meaning that the smaller the effect size, the larger the total sample size required to detect the effect.

Based on this statistical decision chart, which statistical tests are used for unrelated designs across Nominal, Ordinal, and Interval levels of measurement?

For unrelated difference designs: Nominal data uses Chi-Squared; Ordinal data uses Mann-Whitney U; Interval data uses Unrelated t-test.

Which statistical tests should be chosen to evaluate a relationship between co-variables for Ordinal versus Interval data?

Ordinal data uses Spearman's rho, while Interval data uses Pearson's $r$.

How do positive, negative, and zero skewness values describe a distribution?

A skewness of $0$ means the distribution is symmetrical or normal. Negative skewness means the data clusters to the right (tail stretches left). Positive skewness means the data clusters to the left (tail stretches right).

What percentages of data fall within one, two, and three standard deviations of the mean in a normal distribution?

$68.3\%$ falls within $\mu \pm \sigma$, $95.5\%$ falls within $\mu \pm 2\sigma$, and $99.7\%$ falls within $\mu \pm 3\sigma$.

What is a spurious relationship in statistics?

A situation where two variables appear to have an association (e.g., height and maths performance), but the association disappears when a third grouping variable (such as grade level) is taken into account.

What is Simpson's paradox?

A phenomenon where an association between two variables vanishes or entirely reverses direction when a grouping variable is applied.

What measurement problem is shown in this distribution of exam scores, and how does it affect variance and the mean?

It illustrates a ceiling effect (scores clustering at the maximum limit of $100$). It reduces the variance of the variable and distorts the mean, making it higher than it would be if the test were unbiased.

What is a floor effect, and what factors can cause it?

A floor effect occurs when scores cluster at the minimum possible value. It indicates that a measure or exam is excessively difficult, or reflects a desirability bias.

How do linear transformations of adding a constant ($X' = X + b$) and multiplying by a constant ($X' = aX$) affect the Mean ($M$) and Standard Deviation ($S$)?

Adding constant $b$: Mean becomes $M + b$, while $SD$ remains unchanged ($S$). Multiplying by constant $a$: Mean becomes $M \times a$, and $SD$ becomes $S \times a$.

What is the formula and definition of a z-score?

A z-score (standard score) measures how far and in what direction a score lies from the mean in standard deviation units. It is calculated as: $z = \frac{X - \bar{X}}{S}$, where $X$ is the score, $\bar{X}$ is the mean, and $S$ is the standard deviation.

Can standardisation convert a non-normally distributed variable into a normally distributed one?

No. Standardisation is a linear transformation that only rescales and shifts data without altering the shape; if data is not normally distributed before standardisation, it will not be normally distributed after.

What are four common non-linear transformations, and why are they used?

Common transformations include: Square root ($X' = \sqrt{X}$), Inverse/reciprocal ($X' = \frac{1}{X}$), Power ($X' = X^2$), and Log ($X' = \log_{10}(X)$). They are used to alter distribution shape, make data easier to understand, reduce skew, minimise outliers, and linearise relationships.