AP Statistics: Inference for Quantitative Data (Means)

Normal (z) distribution

A probability distribution used when the population standard deviation is known and the sampling distribution can be assumed normal.

Student’s t-distribution

A probability distribution used when the population standard deviation is unknown and is estimated using the sample standard deviation.

Degrees of Freedom (df)

A parameter that describes the number of independent values or observations in a statistical calculation, often calculated as n - 1 for a one-sample t-test.

Standard Error (SE)

The estimated standard deviation of the sampling distribution of a statistic, typically the mean.

Confidence Interval (CI)

A range of values that is used to estimate the true population parameter with a certain level of confidence.

Hypothesis Testing

A method of making decisions using data, whether to accept or reject a hypothesis based on sample data.

Null Hypothesis (H0)

A statement of no effect or no difference that is tested to determine if there is enough evidence to reject it.

Alternative Hypothesis (Ha)

The statement that is accepted if the null hypothesis is rejected; it represents an effect or difference.

Central Limit Theorem (CLT)

A statistical theory that states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases.

Random Sample

A sample that is selected in such a way that every member of the population has an equal chance of being included.

10% Condition

A condition that states when sampling without replacement, the sample size must be less than 10% of the population.

Robust Procedure

A statistical method that produces valid results even when certain assumptions are violated.

t-statistic

A ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error.

P-value

The probability of observing a test statistic as extreme as, or more extreme than, the value observed under the null hypothesis.

Power of a Test

The probability of correctly rejecting a false null hypothesis, equivalent to 1 minus the probability of a Type II error.

Type I Error

The error made when a true null hypothesis is rejected; also known as a false positive.

Type II Error

The error made when a false null hypothesis is not rejected; also known as a false negative.

Standardized Score (z-score)

A measure that describes a value's relation to the mean of a group of values, indicating how many standard deviations away the value is.

t-distribution Shape

Bell-shaped and symmetric around zero, but has fatter tails than the standard normal distribution.

Assumptions for Inference

Three conditions: Random sample, 10% condition, and Normal/Large sample requirement that must be verified before performing analysis.

Degrees of Freedom for One Sample

Calculated as n - 1, used in t-tests to account for sample size.

Point Estimate

A single value estimate of a population parameter, such as the sample mean as an estimate of the population mean.

Critical Value

The value that a test statistic must exceed in order to reject the null hypothesis, determined by a chosen significance level.

t* Value

The critical value from the t-distribution corresponding to the desired confidence level.

Paired Data

Data collected from the same subject at two different times or under two different conditions.

Independent Samples

Two groups of data that have no relationship to each other.

Simulation for P-values

Estimating a P-value using repeated random sampling under the null hypothesis to observe the relative frequency of extreme results.

Statistical Significance

A result that is unlikely to have occurred by chance, typically determined by comparing the p-value to a preset alpha level.

Effect Size

A quantitative measure of the magnitude of the experimental effect; larger effect sizes increase the power of a test.

Confidence Level

The frequency (expressed as a percentage) at which the confidence interval will capture the true parameter value across many samples.

Fail to Reject H0

The conclusion drawn when there is not enough evidence to reject the null hypothesis; does not imply H0 is true.

H0: BC = BC0

The null hypothesis stating that the population mean is equal to a specific value.

Mean of Differences (μdiff)

In paired data, the average of the differences between paired observations.

Obtain P-value

The process of calculating the p-value based on the t-statistic and degrees of freedom in hypothesis testing.

Confidence Interval Interpretation

An interval estimate that reflects our confidence that it contains the true population parameter.

Satterthwaite approximation

A method for calculating degrees of freedom for comparing two means that provides a more nuanced estimate.

P-value Approximation

Using relative frequency from simulations to estimate the likelihood of observing a test statistic as extreme as the one from the sample.

Estimate Standard Deviation of the Sampling Distribution

Using the sample standard deviation to calculate the standard error.

Independent Random Samples

Random samples taken from two distinct populations where each sample does not influence the other.

Chi-Squared Distribution

A distribution that contains all the distributions for the test of independence; often used for testing relationships between variables.

Assumption of Normality

The assumption that a population is normally distributed; particularly important for small sample sizes.

M (Make Decision) in PHANTOMS

The step in the PHANTOMS method that involves determining whether to reject or fail to reject the null hypothesis based on the p-value.