7ª AULA - BIOESTATÍSTICA E EPIDEMIOLOGIA - 52/25

Main ideas / lessons

• Course focus for this lesson: statistical inference in biostatistics/epidemiology, specifically hypothesis testing.
• Transition in the course: after finishing the cycle of descriptive + inferential statistics, the next classes move into epidemiology.
  • Units 5, 6, 7 are mainly theoretical.
  • The next class emphasizes indicator calculations.
• Why hypothesis tests matter in health/epidemiology: they support decision-making about population parameters (e.g., whether a disease behaves differently, whether a machine/medicine is within acceptable targets) using evidence from a sample.

Core methodology: Hypothesis testing framework (as taught)

1) Data/background assumptions

• A population is characterized by a parameter (often the population mean in this lesson).
• A sample is drawn using sampling techniques; measurements can be quantitative/qualitative.
• The lesson mostly uses Z tests for means when population variance/SD is known and/or the scenario matches normal/Z-test conditions.
  • Later examples still reference table-choice decisions.

2) Formulate hypotheses

Always set up:

• Null hypothesis (H0): includes an equality sign.
• Alternative hypothesis (H1): is the complement of H0.

Test direction choices:

• Two-tailed (bilateral) test: reject H0 if the sample mean is too high or too low.
  • H0: population mean equals the reference mean
  • H1: mean is different
• Right-tailed (unilateral) test:
  • H0: population mean ≤ reference
  • H1: population mean > reference
• Left-tailed (unilateral) test:
  • H0: population mean ≥ reference
  • H1: population mean < reference

3) Choose significance level and interpret confidence

• Significance level (α):
  • probability of making a Type I error (rejecting H0 when H0 is true).
• Confidence level:
  • typically 1 − α (e.g., α = 0.05 → confidence = 0.95).
• Emphasis in class:
  • α is “how much you accept being wrong”
  • confidence is “how much you want to be right”

4) Types of errors

• Type I error (α):
  • reject H0 even though H0 is true.
• Type II error (β):
  • fail to reject H0 even though H0 is false.
• Relationship noted:
  • probability of being correct when H0 is true is 1 − α.
• A “table”/decision diagram is mentioned as useful later in epidemiology.

5) Compute the test statistic (Z or critical Z concept)

For the mean (Z framework), the taught statistic is:

[ Z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} ]

where:

• (\bar{x}) = sample mean
• (\mu_0) = population/reference mean under H0
• (\sigma) = population standard deviation
• (n) = sample size

Interpretation is done via critical regions on the normal curve.

6) Decision rule using critical region(s)

• Determine critical Z thresholds from the normal table for the chosen α and tail type.
• Example threshold:
  • for 95% confidence in a two-tailed Z test: critical |Z| = 1.96 (α/2 on each side)
• Compare:
  • If calculated (Z) is in the critical region → reject H0 and accept H1
  • If calculated (Z) is in the non-rejection region → do not reject H0

Examples and applied computations (from the subtitles)

Example 1: Drug filling calibration (bilateral Z test)

Problem setup

• Machine calibrated to deliver 160 ml
• Known standard deviation: σ = 8 ml
• Stop/recalibrate if output is critically high or low
• Sample size: n = 30
• Observed sample mean: (\bar{x}) = 158.20 ml
• Significance level: α = 5% (confidence 95%)
• Use a two-tailed (bilateral) test

Hypotheses

• H0: (\mu = 160)
• H1: (\mu \neq 160)

Test statistic

[ Z = \frac{158.20 - 160}{8/\sqrt{30}} ]

Subtitles result: (Z \approx -1.23)

Critical value / region

• For 95% confidence (two-tailed): critical bounds at ±1.96
• Conclusion (from subtitles):
  • −1.23 is in the non-critical (yellow) region
  • Do not reject H0
  • Decision: no need to stop production line; equipment appears calibrated (with 95% certainty)

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Example 2: Soft drink volume and finding α (two-tailed)

Problem setup

• Label indicates nominal content 290 ml
• Volume (X) is normally distributed with:
  • mean (\mu = 290)
  • variance 36, so σ = 6
• Weekly sample size: n = 9
• Given acceptable interval (from a confidence interval):
  • lower bound: 285.66 ml
  • upper bound: 294.34 ml
• Task: determine significance level α from the given bounds

Hypotheses

• H0: (\mu = 290)
• H1: (\mu \neq 290) (two-tailed)

Table choice explanation

They use Z/normal table because the population variance/SD is known.

Compute critical Z for each bound

For lower bound 285.66:

[ Z_c = \frac{285.66 - 290}{6/\sqrt{9}} = \frac{-4.34}{2} \approx -2.17 ]

For upper bound 294.34:

• (Z_c \approx +2.17)

Convert to confidence / α

• Z-table area extraction in subtitles: 0.4850 (one-side area as described)
• Probability of being “right” computed as:
  • 0.4850 × 2 = 0.97
• Therefore:
  • confidence ≈ 97%
  • α = 3%

Decision implication (as stated)

• With 97% confidence, the interval ([285.66, 294.34]) corresponds to (\mu = 290)
• Therefore “no shutdown necessary” under that interpretation

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Example 3: Multiple-choice style statements using a given CI

Problem setup

• Given: 15 ± 3 represents the 95% confidence interval for the mean
• From a simple random sample size: n = 400
• Significance level stated: 5% (since 95% CI)

They evaluate items (1–3):

Item 1 (evaluated as correct)

• Hypotheses:
  • H0: (\mu = 17)
  • H1: (\mu \neq 17)
• Since 17 lies inside the 95% CI / non-rejection region:
  • no statistical evidence to reject H0
  • statement correct

Item 2 (evaluated as wrong)

• The statement uses a different confidence level (99.9%).
• Argument given:
  • higher confidence → smaller α → interval should expand (larger margin of error)
  • the item’s direction/magnitude is inconsistent
• Conclusion: item 2 is wrong

Item 3 (evaluated as wrong)

• Claim: population standard deviation is 3
• They compute the implied standard deviation using:
  • margin of error formula with Z = 1.96 (95%) and n = 400
• Their calculation yields ≈ 30.61, not 3
• Conclusion: item 3 is wrong

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Examples 4 and 5

• Marked as homework
  • Example 4: Type I vs Type II errors (theoretical)
  • Example 5: similar practice

Additional teaching / logistics

• Reminder to use Word’s equation editor for proper symbols (e.g., (\bar{x}), Greek letters).
• Book recommendation: “The Danger of Being Lucid” (Rosa Monteiro), mentioned as relevant particularly for normal distribution concepts (and psychology-related learning).
• Clarification about symmetry in two-tailed tests:
  • if results differ, look up both sides in the Z table and combine probabilities; symmetry often makes them equal.

Speakers / sources featured

• Professor (main instructor) (name not fully specified in subtitles)
• Professor Ricardo
• Professor Adriana
• Professor Débora
• “Statisticians” (referenced as a general source)
• Book author: Rosa Monteiro (“The Danger of Being Lucid”)