flashcard 6

Question 1

What is the primary goal of inferential statistics?

Answer 1

To draw conclusions about a population based on sample data by testing hypotheses and estimating the probability that observed patterns arose by chance.

Question 2

List the main steps involved in performing a formal statistical test.

Answer 2

1) Formulate null (H₀) and alternative (H₁) hypotheses; 2) Choose an appropriate test based on data type and study design; 3) Compute the test statistic and associated P-value; 4) Evaluate statistical significance; 5) Reject or fail to reject H₀.

Question 3

How is a null hypothesis (H₀) defined, and what role does it play in hypothesis testing?

Answer 3

The null hypothesis is a statement assumed true unless data provide sufficient evidence to reject it; it serves as the baseline for comparison, such as “no effect” or “no difference” in the population.

Question 4

What distinguishes a one-tailed test from a two-tailed test?

Answer 4

A one-tailed test examines an effect in a specified direction (e.g., increase only), while a two-tailed test considers deviations in either direction (increase or decrease). The choice depends on prior knowledge or logical constraints indicating which direction is plausible.

Question 5

Why might a researcher choose a one-tailed test instead of a two-tailed test?

Answer 5

When previous literature, biological constraints, or physical logic indicate that any true effect can only go in one direction, making a one-tailed test more powerful for detecting that specific directional change.

Question 6

What does the P-value represent in hypothesis testing?

Answer 6

The P-value is the probability of obtaining the observed result— or one more extreme— assuming that the null hypothesis is true; it quantifies how surprising the data are under H₀.

Question 7

How is statistical significance conventionally determined using the P-value?

Answer 7

If P < 0.05, the result is typically deemed statistically significant; P < 0.01 or P < 0.001 are considered very or highly significant, respectively. Values between 0.05 and 0.10 are often called “borderline significant,” suggesting that more data might clarify the effect.

Question 8

What is a Type I error, and how is it related to the significance level (α)?

Answer 8

A Type I error occurs when one incorrectly rejects a true null hypothesis (false positive). The significance level α (e.g., 0.05) is the maximum tolerated probability of making that error.

Question 9

What is a Type II error, and what concept quantifies our ability to avoid it?

Answer 9

A Type II error happens when one fails to reject a false null hypothesis (false negative). Statistical power (1 − β) measures the probability of correctly detecting a true effect, thus avoiding Type II errors.

Question 10

Why is it incorrect to say a large P-value proves there is no effect?

Answer 10

A large P-value only indicates insufficient evidence to reject H₀; it does not prove that no effect exists, as lack of evidence could stem from small sample size or high variability rather than a true null condition.

Question 11

When testing for a linear relationship between two continuous variables, which correlation coefficient is appropriate?

Answer 11

Pearson’s correlation coefficient is used when both variables are continuous, normally distributed, and expected to have a linear relationship.

Question 12

Under what circumstances should Spearman’s rank correlation be used instead of Pearson’s?

Answer 12

When data are ordinal, skewed, not normally distributed, or the relationship is monotonic but not strictly linear; Spearman’s assesses correlation based on ranks and is robust to outliers.

Question 13

What does a Pearson correlation coefficient (r) of zero indicate, and why does it not necessarily imply no association?

Answer 13

r = 0 indicates no linear relationship between variables, but a non-linear or curvilinear association may still exist; thus, zero linear correlation doesn’t guarantee complete independence.

Question 14

What is a chi-square (χ²) test, and when is it applied?

Answer 14

The χ² test evaluates whether observed frequencies in categorical data deviate from expected frequencies under the assumption of no association; it’s used for contingency tables to test independence or goodness of fit.

Question 15

How are expected frequencies calculated for a simple 2×2 contingency table in the χ² test?

Answer 15

For each cell, multiply the row total by the column total, then divide by the overall sample size; this gives the expected count if there is no association between row and column variables.

Question 16

Why might Fisher’s exact test be used instead of a χ² test?

Answer 16

Fisher’s exact test provides an exact P-value for small sample sizes (when one or more expected counts are < 5), whereas the χ² test is an approximation that can be inaccurate under those conditions.

Question 17

What assumptions underlie Student’s t-test for comparing two means?

Answer 17

Data in each group should be approximately normally distributed, groups should have equal variances (for the classic t-test), and observations must be independent. If variances are unequal, Welch’s t-test is preferred.

Question 18

When should an unpaired t-test be used over a paired t-test?

Answer 18

Use an unpaired t-test for independent samples (e.g., two different groups of subjects). A paired t-test is for dependent samples or repeated measures (e.g., the same subjects measured before and after treatment).

Question 19

How can one verify whether data meet the normality assumption before applying a parametric test?

Answer 19

By visual inspection of histograms or Q-Q plots, or via formal tests like the D’Agostino–Pearson normality test; if data deviate substantially from normality, nonparametric alternatives should be considered.

Question 20

What is the principle behind the Mann–Whitney U-test?

Answer 20

The Mann–Whitney U-test (Wilcoxon rank-sum) compares the ranks of two independent samples to test whether one group tends to have larger values than the other; it does not assume normality.

Question 21

Under what conditions is the Mann–Whitney U-test appropriate?

Answer 21

When comparing two independent groups with continuous or ordinal data that are not normally distributed, provided the distributions have similar shapes and sample sizes are adequate.

Question 22

What does ANOVA (Analysis of Variance) test, and why is it used instead of multiple t-tests?

Answer 22

One-way ANOVA tests whether there are any statistically significant differences among the means of three or more independent groups; it controls the family-wise error rate better than performing multiple pairwise t-tests.

Question 23

What assumption does ANOVA make about the variances across groups, and what alternative exists if this assumption is violated?

Answer 23

ANOVA assumes homogeneity of variances (equal variances across groups). If variances are unequal, alternatives include Welch’s ANOVA or, for non-normal data, the Kruskal–Wallis test.

Question 24

Explain the difference between one-way and two-way ANOVA.

Answer 24

One-way ANOVA examines the effect of a single categorical predictor on a continuous outcome. Two-way ANOVA evaluates the effects of two categorical predictors (and their interaction) on a continuous outcome.

Question 25

Why are post-hoc tests necessary following a significant ANOVA result?

Answer 25

ANOVA only indicates that at least two group means differ, but doesn’t specify which ones; post-hoc tests (e.g., Tukey’s HSD) perform pairwise comparisons with adjustments to control the overall Type I error rate.

Question 26

What is the Tukey Honest Significant Difference (HSD) method used for?

Answer 26

To conduct all pairwise comparisons among group means after a significant one-way ANOVA, providing adjusted P-values that control the family-wise error rate under the assumption of equal variances.

Question 27

When might the Games–Howell post-hoc test be preferred over Tukey’s HSD?

Answer 27

When group variances are unequal—an assumption of Tukey’s HSD—because Games–Howell does not require homogeneity of variances to perform pairwise comparisons.

Question 28

What is the Kruskal–Wallis test, and when should it be used?

Answer 28

The Kruskal–Wallis test is a nonparametric alternative to one-way ANOVA that compares the ranks of multiple independent groups; it’s used when data are not normally distributed or when variances differ substantially.

Question 29

How does the Kruskal–Wallis test handle ties and unequal sample sizes?

Answer 29

It ranks all observations jointly, accounting for ties by assigning average ranks to tied values; unequal sample sizes are handled naturally since the test compares overall rank sums relative to group sizes.

Question 30

Define statistical power and explain two ways to increase it.

Answer 30

Statistical power is the probability of correctly rejecting a false null hypothesis (1 − β). It can be increased by enlarging the sample size or reducing data variability (e.g., using more precise measurements).

Question 31

What is the difference between statistical significance and practical (or clinical) significance?

Answer 31

Statistical significance refers to the likelihood that an observed effect is not due to chance. Practical significance considers the real-world importance or magnitude of that effect—whether it is large enough to matter.

Question 32

What is a confidence interval, and how should it be interpreted?

Answer 32

A confidence interval provides a range of plausible values for a population parameter (e.g., a mean) with a specified confidence level (e.g., 95%). It means that, across repeated samples, the true parameter will lie within that range 95% of the time.

Question 33

Why does a large sample size sometimes lead to “statistically significant” results that might be misleading?

Answer 33

With very large samples, even trivial differences can produce very small P-values, making them statistically significant despite having negligible practical impact.

Question 34

Explain the concept of “degrees of freedom” in the context of the χ² test.

Answer 34

Degrees of freedom for a contingency table are calculated as (number of rows − 1) × (number of columns − 1). They reflect how many cell counts can vary independently when computing expected frequencies.

Question 35

How do researchers assess if two categorical variables are independent?

Answer 35

By constructing a contingency table and using a χ² test of independence (or Fisher’s exact test if cell counts are small) to see if observed frequencies deviate significantly from those expected under independence.

Question 36

In hypothesis testing, what does “rejecting the null hypothesis” imply?

Answer 36

That there is sufficient statistical evidence to conclude that the null hypothesis (e.g., “no effect” or “no difference”) is unlikely to be true, supporting the alternative hypothesis.

Question 37

Describe one scenario in which a researcher would use a two-sided alternative hypothesis.

Answer 37

When there is no prior indication of whether the effect could be positive or negative—only that it differs from zero—so the test must detect deviations in either direction.

Question 38

Why is formulating the alternative hypothesis crucial before analyzing data?

Answer 38

Because the specific wording of H₁ determines whether a one-tailed or two-tailed test is appropriate and ensures that the analysis aligns with the research question and prior knowledge.

Question 39

What is the purpose of testing for normality in quantitative data?

Answer 39

To determine whether parametric tests (which assume normal distribution) are valid or if nonparametric alternatives should be used; normality tests and plots help identify distributional deviations.

Question 40

Name two formal tests for assessing normality and briefly describe how they differ.

Answer 40

1) D’Agostino–Pearson test, which combines skewness and kurtosis measures into a single P-value; 2) Shapiro–Wilk test, which assesses how well data fit a normal distribution based on correlation with theoretical normal scores.

Question 41

When is Welch’s t-test preferred over the classic Student’s t-test?

Answer 41

When comparing two independent samples that do not have equal variances, as Welch’s adjusts the degrees of freedom to account for unequal variability.

Question 42

How does the concept of ranking underlie many nonparametric tests?

Answer 42

Nonparametric tests often convert raw data into ranks, then compare rank sums or average ranks across groups, minimizing the influence of outliers and not requiring normality.

Question 43

What does it mean for two variables to have a monotonic relationship?

Answer 43

As one variable increases, the other either consistently increases or consistently decreases, but not necessarily at a constant rate; Spearman’s correlation captures monotonic associations.

Question 44

Explain how sample size influences both the standard error of the mean and the P-value.

Answer 44

Larger sample sizes reduce the standard error of the mean, leading to narrower confidence intervals and increasing the likelihood of detecting statistically significant differences (smaller P-values), all else equal.

Question 45

What is the rationale for using a two-way ANOVA with interaction?

Answer 45

To test not only the main effects of two factors on an outcome but also whether the effect of one factor depends on the level of the other factor, revealing synergistic or antagonistic interactions.

Question 46

Under what condition would you choose the Kruskal–Wallis test over one-way ANOVA?

Answer 46

When comparing more than two independent groups whose data violate normality assumptions or have heterogeneity of variances, making ANOVA inappropriate.

Question 47

Why is it important to ensure sample independence when selecting a statistical test?

Answer 47

Because many tests (e.g., unpaired t-test, χ² test, one-way ANOVA) assume that observations are independent; violating this can inflate Type I error rates and invalidate results.

Question 48

Summarize the conceptual difference between descriptive and inferential statistics.

Answer 48

Descriptive statistics summarize and visualize data from a specific sample (e.g., means, medians, histograms), while inferential statistics use sample data to make generalizations, test hypotheses, and estimate population parameters.

Question 49

What key information must be reported alongside a P-value to provide proper context?

Answer 49

The test statistic, sample size, direction of the effect (if one-tailed), confidence intervals for estimates, and an indication of practical or clinical significance to ensure transparent interpretation.

Question 50

How does family-wise error rate relate to multiple comparisons, and how do post-hoc tests address it?

Answer 50

Family-wise error rate is the probability of making at least one Type I error across all pairwise comparisons. Post-hoc methods (e.g., Tukey’s HSD) adjust P-values or significance thresholds to control this overall error rate when comparing multiple groups.