ISDS 2000 Unit 3: Confidence Intervals and Hypothesis Testing

Vocabulary flashcards covering core concepts, symbols, formulas, hypothesis testing procedures, condition checks, and Excel auditing rules from ISDS 2000 Unit 3.

Population

The whole group of individuals or items that an investigator wants to understand.

Sample

The subset of a population that is actually observed and measured.

Parameter

A numerical characteristic of a population, such as the population mean ($\mu$) or population proportion ($p$).

Statistic

A numerical summary computed from sample data, such as the sample mean ($\bar{x}$) or sample proportion ($\hat{p}$), used to estimate a population parameter.

Standard Deviation

A numerical measure describing variability among individual observations in a dataset.

Standard Error ($SE$)

A numerical measure describing the sampling variability of a statistic across repeated samples, calculated by dividing standard deviation by $\sqrt{n}$.

Margin of Error ($E$)

The half-width of a confidence interval, calculated as a critical value multiplied by the standard error ($E = \text{critical value} \times SE$).

Degrees of Freedom ($df$)

A parameter characterizing the $t$ distribution; for a one-mean $t$ procedure, it equals $n - 1$.

One-Mean $z$ Confidence Interval

An interval estimation method for a population mean ($\mu$) given by $\bar{x} \pm z^* \times \left(\frac{\sigma}{\sqrt{n}}\right)$, used when the population standard deviation ($\sigma$) is known.

One-Mean $t$ Confidence Interval

An interval estimation method for a population mean ($\mu$) given by $\bar{x} \pm t^* \times \left(\frac{s}{\sqrt{n}}\right)$ with $df = n - 1$, used when the population standard deviation ($\sigma$) is unknown and estimated by $s$.

One-Proportion $z$ Confidence Interval

An interval estimation method for a binary population proportion ($p$) given by $\hat{p} \pm z^* \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $\hat{p} = \frac{x}{n}$.

Interpretation of 95% Confidence

The long-run property of the procedure stating that if random samples were repeatedly drawn and intervals constructed using the same method, approximately $95\%$ of those intervals would capture the fixed population parameter.

Conservative Planning Value ($p^*$)

The planning proportion $p^* = 0.50$ used when no reliable prior estimate is available, which maximizes $p^*(1 - p^*) = 0.25$ to yield the largest required sample size.

Proportion Sample Size Formula

The formula $n = \frac{(z^*)^2 p^*(1 - p^*)}{E^2}$, used to determine the minimum number of observations required to achieve a desired margin of error $E$, where any fractional result must be rounded up.

Majority Support Criterion

The decision standard stating that sample evidence demonstrates majority support only when the entire confidence interval lies strictly above $50\%$.

Nonresponse Bias

A nonsampling survey bias occurring when people who choose to respond differ systematically from those who do not, which is not eliminated or measured by the margin of error.

Coverage Bias

A nonsampling survey bias occurring when the sampling frame systematically misses or excludes part of the target population.

Null Hypothesis ($H_0$)

The baseline statement specifying no effect, no change, or equality to a benchmark parameter ($\mu = \mu_0$ or $p = p_0$), used to perform test calculations.

Alternative Hypothesis ($H_a$)

The claim specifying a departure from the null benchmark in a particular direction (less than, greater than, or different from) that the test seeks evidence to support.

P-value

The probability, calculated under the assumption that the null hypothesis ($H_0$) is true, of obtaining a test statistic as extreme as or more extreme than the observed statistic in the direction(s) specified by $H_a$.

Significance Level ($\alpha$)

The predetermined threshold for Type I error probability; the null hypothesis is rejected whenever the p-value satisfies $p \le \alpha$.

Type I Error

The error of rejecting the null hypothesis when the null hypothesis is actually true (a false alarm).

Type II Error

The error of failing to reject the null hypothesis when the null hypothesis is actually false (a missed real effect).

Statistical Power ($1 - \beta$)

The probability of correctly rejecting the null hypothesis when a real effect or difference exists ($1 - \beta$, where $\beta$ is the Type II error probability).

One-Mean $z$ Test

A hypothesis test for a population mean using test statistic $z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$, used when the population standard deviation ($\sigma$) is known.

One-Mean $t$ Test

A hypothesis test for a population mean using test statistic $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$ with $df = n - 1$, used when the population standard deviation ($\sigma$) is unknown and estimated by $s$.

One-Proportion $z$ Test

A hypothesis test for a population proportion using test statistic $z = \frac{\hat{p} - p_0}{SE_0}$, where $SE_0 = \sqrt{\frac{p_0(1 - p_0)}{n}}$.

Null Standard Error ($SE_0$)

The standard error used specifically in a one-proportion $z$ test, computed using the hypothesized null proportion: $SE_0 = \sqrt{\frac{p_0(1 - p_0)}{n}}$.

Observed Counts Condition

The sample size check for a one-proportion confidence interval requiring that observed successes ($x = n\hat{p}$) and observed failures ($n - x = n(1 - \hat{p})$) both be at least $5$.

Expected Counts Condition

The sample size check for a one-proportion $z$ test requiring that expected successes ($np_0$) and expected failures ($n(1 - p_0)$) both be at least $5$.

Matched Two-Sided Test and Interval Rule

The equivalence rule stating that for matching levels ($C = 1 - \alpha$), a null benchmark value ($\mu_0$) is rejected in a two-sided test if and only if it falls outside the matching two-sided confidence interval.

Excel NORM.S.INV

The Excel function that takes a cumulative left-tail probability and returns the corresponding standard normal critical cutoff ($z^*$).

Excel T.INV.2T

The Excel function that takes a two-tailed probability ($\alpha$) and degrees of freedom ($df$) to return the positive critical cutoff ($t^*$).

Excel NORM.S.DIST

The Excel function that accepts an observed $z$ statistic and TRUE to return the cumulative probability to its left.

Excel T.DIST.RT

The Excel function that accepts an observed $t$ statistic and degrees of freedom to return the right-tail p-value area.

Excel T.DIST.2T

The Excel function that accepts the absolute magnitude of an observed $t$ statistic (ABS(t)) and degrees of freedom to return the two-tailed p-value.