biostats exam 2

Hypothesis testing

Uses sample data to make inferences about a population and test whether data are compatible with a specific claim about the population.

Estimation vs hypothesis testing

Estimation asks how large an effect is while hypothesis testing asks whether there is any effect at all.

Null hypothesis (H0)

Statement about a population parameter that assumes no effect, no difference, or no relationship; assumed true at the start of testing.

Alternative hypothesis (HA)

Includes all other possible values of the parameter besides the value stated in the null hypothesis.

Examples of alternative hypotheses

p ≠ 0.5, p > 0.5, p < 0.5.

Hypothesis testing steps

1) State hypotheses 2) Compute test statistic 3) Determine p-value 4) Draw conclusion.

Two-sided test

Tests if the parameter differs from the null value in either direction.

One-sided test

Tests if the parameter differs from the null value in only one direction.

Test statistic

Calculated value from sample data that measures how much the data differ from the null hypothesis.

Null distribution

Probability distribution of the test statistic assuming the null hypothesis is true.

P-value

Probability of obtaining results as extreme or more extreme than observed assuming the null hypothesis is true.

Interpretation of small p-value

Data are unlikely under the null hypothesis and provide evidence against it.

Significance level (α)

Probability threshold used to decide whether to reject the null hypothesis (commonly 0.05).

Decision rule using p-value

If p ≤ α reject H0; if p > α fail to reject H0.

Power of a statistical test

Probability of rejecting a false null hypothesis.

Factors affecting statistical power

Sample size, effect size, variability, and significance level.

Type I error

Rejecting a true null hypothesis.

Probability of Type I error

Equal to the significance level α.

Reducing Type I error

Use a smaller significance level (e.g., 0.01 instead of 0.05).

Type II error

Failing to reject a false null hypothesis.

Relationship between Type II error and power

Lower Type II error corresponds to higher statistical power.

Binomial distribution

Probability distribution describing number of successes in a fixed number of independent trials with constant probability of success.

Binomial distribution parameters

n = number of trials, x = number of successes, p = probability of success.

Binomial distribution formula

P(x) = (n choose x) p^x (1−p)^(n−x).

Expected value of binomial distribution

E(x) = np.

Null expectation example

If n = 18 and p = 0.5 then expected successes = 9.

Two-sided binomial p-value calculation

Probability of observed value and more extreme values on both tails.

Confidence interval

Range of values likely to contain the true population parameter.

Relationship between confidence intervals and hypothesis testing

If the null value lies outside the confidence interval reject H0.

Proportion estimate

p̂ = x/n where x = number in category and n = total observations.

Standard error of proportion

SE = √[p̂(1−p̂)/n].

Agresti-Coull adjusted proportion

p′ = (x+2)/(n+4).

Interpretation of CI significance

If the null value lies outside the confidence interval the result is statistically significant.

Categorical data

Qualitative characteristics describing group membership rather than magnitude.

Nominal variables

Categories without inherent order.

Ordinal variables

Categories with natural ranking or order.

Observational study

Researcher does not assign treatments and simply observes exposures or outcomes.

Cross-sectional study

Observes a population at a single point in time.

Longitudinal study

Observes the same individuals repeatedly over time.

Case-control study

Compares individuals with a disease to controls and looks backward for exposures.

Experimental study

Researcher assigns treatments to participants.

Randomized controlled trial

Participants randomly assigned to treatment or control groups.

2×2 contingency table

Table used to analyze relationship between two categorical variables with two categories each.

Relative risk (RR)

Ratio of probability of an outcome in exposed group to probability in control group.

Risk formula

Risk = number of new cases / population at risk.

Relative risk formula

RR = (a/(a+b)) / (c/(c+d)).

Relative risk interpretation RR=1

No association between exposure and outcome.

Relative risk interpretation RR>1

Exposure increases the risk of the outcome.

Relative risk interpretation RR<1

Exposure decreases the risk of the outcome.

Confidence interval interpretation for RR

If CI includes 1 the association is not statistically significant.

Odds

Ratio of probability of success to probability of failure.

Odds formula

Odds = p/(1−p).

Odds ratio (OR)

Compares odds of an outcome between two groups.

Odds ratio shortcut formula

OR = (a×d)/(b×c).

Odds ratio interpretation OR=1

No association between exposure and outcome.

Odds ratio interpretation OR>1

Outcome more likely in exposed group.

Odds ratio interpretation OR<1

Outcome less likely in exposed group.

When to use odds ratio

Case-control studies, rare outcomes in cross-sectional studies, and logistic regression.

Chi-square goodness-of-fit test

Compares observed frequency distribution with expected distribution under a probability model.

Limitation of binomial test

Only works with two mutually exclusive outcomes.

Advantage of chi-square goodness-of-fit test

Can analyze more than two categories.

Expected frequency formula (goodness-of-fit)

Expected = probability × sample size.

Chi-square statistic formula

χ² = Σ (O − E)² / E.

Degrees of freedom for goodness-of-fit test

df = number of categories − 1.

Decision rule for chi-square test

If χ² > critical value reject H0; if χ² ≤ critical value fail to reject H0.

Chi-square contingency test

Tests whether two categorical variables are independent.

Expected frequency formula (contingency table)

E = (row total × column total) / grand total.

Degrees of freedom for contingency table

df = (rows−1)(columns−1).

Assumptions for chi-square tests

No expected frequency less than 1 and no more than 20% of categories with expected frequency less than 5.

Interpretation of chi-square test

If χ² exceeds critical value there is a significant association between variables.

Normal distribution

Continuous probability distribution forming a symmetrical bell-shaped curve.

Properties of normal distribution

Symmetrical, single mode, mean = median = mode, total area under curve = 1.

Parameters of normal distribution

μ = mean and σ = standard deviation.

Probability in normal distribution

Measured as area under the curve representing proportion of observations.

Empirical rule

68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD of the mean.

Standard normal distribution

Normal distribution with mean 0 and standard deviation 1.

Z-score

Number of standard deviations a value lies from the mean.

Z-score formula

Z = (X − μ) / σ.

Interpretation of Z=0

Value equals the mean.

Interpretation of Z>0

Value above the mean.

Interpretation of Z<0

Value below the mean.

Sampling distribution

Distribution of a statistic across repeated samples from a population.

Standard error of the mean

SE = σ/√n.

Effect of sample size on SE

Larger sample sizes reduce standard error and increase precision.

Central limit theorem

Distribution of sample means approaches a normal distribution as sample size increases regardless of population distribution.

Normal approximation to binomial

Binomial distribution can be approximated by a normal distribution when sample size is large.

Conditions for normal approximation

np ≥ 5 and n(1−p) ≥ 5.

Mean of binomial distribution

μ = np.

Standard deviation of binomial distribution

σ = √[np(1−p)].

Continuity correction

Adjustment of ±0.5 when approximating discrete binomial distribution with continuous normal distribution.

Continuity correction rule (include value)

P(X ≥ x) → x − 0.5 and P(X ≤ x) → x + 0.5.

Continuity correction rule (exclude value)

P(X > x) → x + 0.5 and P(X < x) → x − 0.5.

Spider example mean

μ = np = 20.5.

Spider example standard deviation

σ ≈ 3.20.

Spider example continuity correction

P(X ≥ 31) becomes 30.5.

Spider example Z-score

Z = (30.5 − 20.5)/3.20 ≈ 3.12.

Spider example probability

P(Z > 3.12) ≈ 0.0009.

Spider example p-value (two-sided)

p = 2 × 0.0009 = 0.0018.

Hypothesis test interpretation rule

If p < α reject H0; if p ≥ α fail to reject H0.