Hypothesis Testing in Biostatistics

Essential Question: In hypothesis testing, we are asking, “Could these observations have occurred by chance?”
- Source: Gonick and Smith (10/8/2025)

Null Hypothesis (H0): This hypothesis states that the observations have occurred purely by chance.
- Source: Gonick and Smith
Scientific Hypothesis (Alternate Hypothesis, HA): This suggests that there is a significant underlying relationship between two variables; for instance, the mean of a random variable may be higher in a particular subgroup.

Historical Context:
- In the 1950s-60s, despite a high population of African American citizens in Southern states, there was a striking absence of African-Americans on jury panels.
- When African American defendants challenged their guilty verdicts, they presented statistical evidence indicating unfair jury selection.
Null Hypothesis for Jury Selection: Jurors are drawn at random from a list of eligible citizens and thus represent the ethnic composition of the population. Any observed deviation is assumed to be due to chance.
Alternate Hypothesis for Jury Selection: There is a disproportionately low number of African-Americans selected for the jury panel, implying potential discrimination.

Proportion Calculation: What is the expected proportion of African-Americans on the jury panel if jurors are selected randomly?
- Population Proportion: 50% of eligible citizens for jury selection were African-American.
Case Scenario: On an 80-person panel of jurors, only 4 were African-American. The question raised is whether this occurrence could be attributed to chance.
Statistical Investigation: If jurors are randomly selected, what are the chances of obtaining a jury with 4 or fewer African-American jurors?

Jury selection modeled as a Binary Random Variable:
- Outcome: Is the juror African-American? (Yes or No)
- Probability of Success (p): $p = rac{1}{2}$ (representing both ethnicities equally)
Binomial Distribution Definition: B(n, p) where "n trials" (number of selected jurors) yield only two outcomes (success or failure).
- Probability of k successes in n trials: Calculated using the formula:
  $P(X=k) = {n ext{ choose } k} imes p^k imes (1-p)^{n-k}$
Random Variable Representation: For this jury case, it is denoted as:
- $X ext{~} B(n=80, p= rac{1}{2})$
- Meaning: X represents the count of African-American jurors in the jury panel.

Specific Question: If the jury panel selection was random, what is the likelihood of having 4 or fewer African-American jurors? The articulated probability expression:
$P(X ext{ ≤ } 4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)$
Expanded Formula Representation:
$P(X ext{ ≤ } 4) = {80 ext{ choose } 0} p^0 (1-p)^{80} + {80 ext{ choose } 1} p^1 (1-p)^{79} + {80 ext{ choose } 2} p^2 (1-p)^{78} + {80 ext{ choose } 3} p^3 (1-p)^{77} + {80 ext{ choose } 4} p^4 (1-p)^{76}$
- Where this would result in a numerical solution indicating extreme unlikelihood leading to useful insights regarding jury selection proficiency.
Result: The calculated probability was approximately 0.0000000000000000014, a figure which indicates a highly unlikely scenario, likened to the rarity of drawing three consecutive royal flushes in poker.

Step 1: Formulate Your Hypotheses
- H0 (Null Hypothesis): The jury was randomly chosen, meaning African-Americans have a probability $p = rac{1}{2}$ of being selected.
- HA (Alternate Hypothesis): The likelihood of African-Americans being selected is less than their proportion in the population, i.e., p < rac{1}{2}.
Step 2: Identify a Test Statistic
- Test Statistic Chosen: Binomial Random Variable $X ext{~} B(80, rac{1}{2})$ .
Step 3: P-Value Determination
- Contextual Meaning: A probability statement answering, “If the null hypothesis is true, what is the probability of observing at least as extreme a test statistic as the one observed?”
- Calculated P-value from binomial distribution: $P(X ext{ ≤ } 4) = 0.0000000000000000014$
Step 4: Compare P-Value to Significance Level Alpha (α)
- Significance level defines the threshold beneath which an effect is considered statistically significant. If $P ext{-value} ext{ ≤ } α$ ,
  - We reject the null hypothesis, suggesting significant deviations in statistics.
- For this case, the expert witness set a stringent alpha of $α = 0.0000000000000000036$ , highlighting the unlikelihood of randomly selected jurors yielding the observed result.
- Conclusion: The judge deemed that the null hypothesis was appropriately rejected, acknowledging irregularities in the jury selection process.

Null Hypothesis (H0): We either "do not reject" or "reject" the null hypothesis.
- It is crucial to note that one never "accepts" the null hypothesis. Instead, statistical tests ascertain if differences arise from sampling variability.
- If data align with H0, one does not reject it. Conversely, if observed data contrasts significantly with H0, it is rejected.
Comparison to Legal Principle: This concept mirrors the legal presumption of innocence where a defendant is deemed innocent (like not rejecting H0) until proven guilty (rejecting H0).

Type I Error: Resulting from rejecting H0 when it is true; associated probability denoted by alpha (α).
- Significance: Setting alpha at a level like 0.05 means that about 5% of the time, we might incorrectly reject the null hypothesis if it indeed is true.
Type II Error: Occurs when failing to reject H0 while it is false, with associated probability denoted by beta (β).
Balancing Errors: Stringent alpha levels reduce Type I error risks but increase the likelihood of Type II errors. Especially in scientific research, there tends to be a preference towards committing Type II errors over Type I errors, with alpha levels usually set at 0.05 or lower for best practices.

Error Awareness in Research Articles: It is crucial to know that if a researcher claims statistical significance at the 0.05 threshold, there's a ~5% chance that they have mistakenly rejected H0 due to a Type I error.
Error Mitigation Strategies: Validating research findings through replication in different populations enhances confidence that results are statistically valid—solution largely lies in replication of studies across cohorts.