biostats exam 2

Hypothesis testing

A statistical method that uses sample data to determine whether a population parameter differs from a specific null expectation.

Estimation vs hypothesis testing

Estimation asks how large an effect is, while hypothesis testing asks whether an effect exists at all.

Null hypothesis (H₀)

A specific statement about a population parameter that assumes no effect, difference, or relationship.

Alternative hypothesis (Hₐ)

A hypothesis that includes all values other than the one specified in the null hypothesis and represents the researcher's claim.

Purpose of the null hypothesis

It serves as the default assumption that is tested using sample data.

Possible decisions in hypothesis testing

Reject the null hypothesis or fail to reject the null hypothesis.

Why we do not say “accept the null hypothesis”

Statistical tests cannot prove the null is true; they only evaluate whether there is enough evidence against it.

Courtroom analogy for hypothesis testing

The null hypothesis is like a defendant being innocent, the alternative hypothesis is guilt, the data are evidence, and the conclusion is guilty or not guilty.

Test statistic

A value calculated from sample data that measures how far the observed result is from what is expected under the null hypothesis.

Example test statistic in the toad study

The number or proportion of right

Expected value under the null hypothesis

The value predicted for the sample statistic if the null hypothesis is true.

Expected value formula

E(X) = n × p.

Sampling error

Natural variation in sample results due to random chance.

Null distribution

The probability distribution of the test statistic assuming the null hypothesis is true.

Purpose of the null distribution

It shows how much variation in results is expected purely from chance.

P

value

Interpretation of a small p

value

Interpretation of a large p

value

Significance level (α)

The threshold probability used to determine whether to reject the null hypothesis.

Common significance level

α = 0.05.

Decision rule using p

value

Two

sided (two

Alternative hypothesis in a two

sided test

One

sided (one

Example of one

sided alternative hypothesis

When one

sided tests are used

Binomial distribution

A probability distribution describing the number of successes in a fixed number of independent trials with the same probability of success.

Conditions for a binomial distribution

Fixed number of trials, two possible outcomes, independent trials, and constant probability of success.

Binomial probability formula

P(X=x) = (n! / (x!(n−x)!)) × p^x × (1−p)^(n−x).

n in the binomial formula

The number of trials.

x in the binomial formula

The number of successes observed.

p in the binomial formula

The probability of success in each trial.

Power of a statistical test

The probability of rejecting a false null hypothesis.

Interpretation of high statistical power

The test is likely to detect a real effect if one exists.

Factors that increase statistical power

Larger sample size, larger effect size, and lower variability.

Type I error

(false negative) Rejecting a true null hypothesis.

Meaning of a Type I error

Concluding there is an effect when there actually is none.

Relationship between α and Type I error

The significance level α sets the probability of making a Type I error.

Reducing Type I error

Using a smaller significance level (for example 0.01 instead of 0.05).

Trade

off when reducing Type I error

Type II error

(false positive) Failing to reject a false null hypothesis.

Meaning of a Type II error

Concluding there is no effect when one actually exists.

Relationship between Type II error and power

A lower probability of Type II error corresponds to higher statistical power.

Goal in hypothesis testing

Minimize both Type I and Type II errors while maintaining adequate statistical power.