Confidence Intervals and Hypothesis Testing

Vocabulary flashcards covering key statistical concepts from the lecture, including confidence intervals, the t-distribution, hypothesis testing formulation, and Type I and Type II errors.

Point Estimate

The center of a confidence interval, denoted by $\bar{x}$ when estimating a population mean and $\hat{p}$ when estimating a population proportion.

Margin of Error

The term added to and subtracted from the point estimate to establish the confidence interval's upper and lower boundaries and account for uncertainty.

Confidence Level

The percentage of intervals that will contain the true population parameter if samples are repeatedly collected and intervals are constructed 100 times.

t-Distribution

A probability distribution comprising a family of curves used when estimating the mean with an unknown standard deviation ($\sigma$); it features fatter tails than the z-distribution and varies with sample size.

Degrees of Freedom

The parameter of the t-distribution that dictates the shape of the distribution curve as the sample size changes.

Standard Error of the Mean

A component of the confidence interval calculated by dividing the sample standard deviation by the square root of the sample size ($\frac{s}{\sqrt{n}}$).

Hypothesis Test

A statistical procedure testing a specific statement or claim regarding an unknown population parameter based on sample data.

Null Hypothesis ($H_0$)

A statement about a population parameter that always contains an equality condition ($=$, $\ge$, or $\le$) and covers the remaining axis complementary to the alternative hypothesis.

Alternative Hypothesis ($H_a$)

The complementary statement to the null hypothesis that does not contain an equal sign and indicates the parameter direction or difference to be supported.

One-Tailed Test

A hypothesis test setup where the critical rejection region lies entirely on one side of the distribution (either strictly smaller or strictly larger than the hypothesized value).

Two-Tailed Test

A hypothesis test with a non-equality alternative ($\neq$) where both an extremely small or an extremely large sample statistic can lead to the rejection of the null hypothesis.

Type I Error

The mistake committed when the null hypothesis ($H_0$) is true in reality, but the decision is made to reject it.

Type II Error

The mistake committed when the null hypothesis ($H_0$) is false in reality, but the test fails to reject it.