BIOL2300 - Exam 2 Content

Chapters 6-9

Hypothesis Testing

Uses sample data to test whether a population parameter differs from a null expectation.

Estimation

Uses sample data to estimate the magnitude or plausible range of a population parameter.

Null Hypothesis (H₀)

The default claim about a population parameter, often no effect, difference, or association.

Alternative hypothesis (H_a)

The claim that a population parameter differs from the null value in a specified way.

Reject H₀

Conclude the sample provides sufficient statistical evidence against the null hypothesis.

Fail to reject H₀

Conclude the sample does not provide sufficient evidence against H₀; it does not prove H₀.

Test Statistic

A value calculated from sample data to assess compatibility with H₀.

Sampling variability

Differences among sample results caused by random sampling, even when H₀ is true.

Null distribution

The sampling distribution of a test statistic when H₀ is assumed true.

P-value

Probability, assuming H₀ is true, of results at least as extreme as those observed.

Significance level (α)

The cutoff used to decide whether a P-value provides enough evidence to reject H₀.

Statistically significant result

A result with P-value ≤ α, leading to rejection of H₀.

Two-sided test

A test with Hₐ allowing values both above and below the null value (≠).

One-sided test

A test with Hₐ allowing departure from the null value in only one direction.

Right-tailed test

A one-sided test with Hₐ: parameter > null value.

Left-tailed test

A one-sided test with Hₐ: parameter < null value.

Binomial hypothesis test

A test of a hypothesized success probability using a fixed number of independent binary trials.

Binomial Distribution

Models the number of successes in n independent trials with constant success probability p.

Expected number of successes

Under the binomial null model, E(X) = n × p.

Statistical power

The probability that a test rejects H₀ when H₀ is false.

Confounding variable

A factor associated with the variables studied that can distort their apparent relationship.

Mutually exclusive outcomes

Outcomes that cannot occur simultaneously in the same trial, such as success and failure.

Independent Trials

Trials in which one outcome does not change the probability of another.

Population proportion (p)

The true fraction of the population belonging to a particular category.

Sample population (p^{^})

The fraction of the sample with the characteristic: p̂ = x/n.

Standard Error (SE)

The estimated typical variation of a sample statistic across repeated samples

95% confidence interval (CI)

An interval from a method that captures the true population parameter in 95% of repeated samples.

Agresti–Coull method

A method for estimating a proportion’s 95% CI using adjusted successes and sample size.

Goodness-of-fit test

Compares observed category frequencies with frequencies predicted by a probability model.

Chi-square (χ²) goodness-of-fit test

Tests whether observed category counts fit the proportions stated in H₀.

Limitation of a binomial test

It handles only two mutually exclusive outcome categories.

Observed frequency (Oᵢ)

The actual number of observations in category i.

Expected frequency (Eᵢ)

The count predicted for category i if the null model is true.

Expected count formula

Eᵢ = n × pᵢ, where pᵢ is the hypothesized category proportion

Degrees of freedom (df)

For a goodness-of-fit test with k fixed category probabilities, df = k−1.

Chi-square null hypothesis

The observed data follow the specified set of population category proportions.

Chi-square alternative hypothesis

At least one population category proportion differs from its specified value.

Large χ² value

Indicates a larger discrepancy between observed and expected counts.

Critical value

The test-statistic threshold marking the rejection region for a chosen α.

Chi-square decision rule

Reject H₀ when the observed χ² exceeds the critical value at the chosen df and α.

Chi-square expected-count condition 1

No category should have an expected frequency less than 1.

Chi-square expected-count condition 2

No more than 20% of categories should have expected frequencies below 5.

Equal-proportion null model

With k equally likely categories, each expected count is n/k.

Categorical data

Qualitative observations organized into categories rather than numeric magnitudes.

Nominal variable

A categorical variable whose categories have no inherent order.

Ordinal variable

A categorical variable whose categories have a natural order.

Contingency analysis

Estimates and tests associations between two or more categorical variables.

2×2 contingency table

A table comparing two categorical variables, each with two categories.

Experimental study

A design in which researchers assign the treatments or exposures.

Observational Study

A design in which researchers observe groups without assigning their exposures.

Randomized Controlled Trial (RCT)

An experiment that randomly assigns participants to treatment and control groups.

Longitudinal Study

A study that follows participants over time to observe later outcomes.

Case–control study

A study selecting cases and controls by outcome, then comparing past exposures.

Exposed group

The group receiving a treatment or having the exposure of interest.

Control or unexposed group

The reference group without the exposure or receiving the comparison condition.

Outcome variable

The result being studied, such as disease, recovery, or an accident.

Incidence

The number of new cases developing in a specified population over a period.

Risk

The proportion of people at risk who develop the outcome.

Relative Risk (RR)

Risk of an outcome in the exposed group divided by risk in the unexposed group.

RR = 1

The outcome risk is the same in both groups.

RR > 1

The outcome risk is higher in the exposed or treatment group.

RR < 1

The outcome risk is lower in the exposed or treatment group.

When to use relative risk

When incidence can be measured, as in longitudinal studies and RCTs.

Odds

The probability of an outcome divided by the probability of no outcome: p/(1−p).

Odds ratio (OR)

Odds of the outcome in one group divided by odds in the comparison group.

OR = 1

Both groups have equal odds of the outcome.

OR > 1

The exposed group has higher odds of the outcome.

OR < 1

The exposed group has lower odds of the outcome.

When to use odds ratio

Commonly used for case–control studies and logistic regression.

OR confidence interval including 1

The interval is compatible with no difference in odds between groups.

RR confidence interval including 1

The interval is compatible with no difference in risk between groups.

Logistic regression

A regression method for categorical outcomes that commonly reports odds ratios.

normal distribution

a continuous probability distribution describing a bell-shaped curve.

• good approximation to the frequency distributions of many biological variables

• two parameters: mean and standard deviation

• symbol Z

Continuous Distribution

A probability distribution in which a variable can take any value within a given range.

Mean (μ)

The average of a population; determines the center of the normal distribution.

Standard Deviation (σ)

Measures how spread out values are around the mean.

Probability Density

Describes how probability is distributed across possible values of a continuous variable.

Area Under the Curve

Represents probability; the total area under a normal distribution equals 1 (100%).

68.3% (~2/3) Rule

Approximately 68.3% of values in a normal distribution fall within one standard deviation of the mean.

95% Rule

Approximately 95% of values fall within two standard deviations of the mean.

Z-probability

the area under the standard normal curve that tells you the chance of a statistic falling below, above, or between specific Z-scores

Z-Score

Measures how many standard deviations an observed value is above or below the mean.

Z-Score Formula

Z = (Y − μ) / σ

• Y is the observed value

• μ is the mean

• σ is the standard deviation.

Positive Z-Score

Indicates that an observed value is above the population mean.

Negative Z-Score

Indicates that an observed value is below the population mean.

Z = 0 meaning

Indicates that the observed value is equal to the population mean.

Standardization

The process of converting observed values into Z-scores to calculate probabilities using the standard normal distribution.

Standard Normal Table (Z-Table)

A table used to find probabilities associated with specific Z-scores.

Right-Tail Probability

The probability that a randomly selected value is greater than a specified Z-score.

Left-Tail Probability

The probability that a randomly selected value is less than a specified Z-score.

Symmetry Rule

The probability below a negative Z-score equals the probability above its corresponding positive Z-score.

Probability Between Two Z-Scores

Calculated by subtracting the probability above the upper bound from the probability above the lower bound.

Z = ±1.96

Approximately 95% of values in a standard normal distribution fall between these Z-scores.

Complement Rule

The probability of an event not occurring equals 1 minus the probability of the event occurring.

Standard Error of the Mean (SE)

The standard deviation of the sampling distribution of sample means.

Effect of Increasing Sample Size

Increasing sample size reduces standard error, making sample means more tightly distributed around the population mean.

Central Limit Theorem (CLT)

States that the distribution of sample means becomes approximately normal with sufficiently large random samples, even when the original population is not normally distributed.

Importance of the Central Limit Theorem

Allows statistical methods that assume normally distributed sample means to be used even when the underlying population is not normal.

Sample Size and CLT

The sample size needed for a good normal approximation depends on the shape of the original population distribution.