Chapters 7, 8, and 9: Sampling Distributions, Confidence Intervals, and Hypothesis Testing

Comprehensive vocabulary flashcards covering sampling distributions, confidence interval construction, sample size calculations, and hypothesis testing procedures from Chapters 7, 8, and 9.

Continuous Variable

A variable that measures a continuous numerical amount where values can include decimals (such as transaction amount, height, weight, or time) and is analyzed using the sample mean $\bar{x}$ to estimate the population mean $\mu$.

Discrete Variable

A variable that represents counts or categorical/yes-no outcomes which cannot be used directly in sample mean methods and must be converted into a sample proportion $\hat{p}$ by dividing the count of interest by the sample size $n$.

Sampling Distribution of the Mean Normality Condition

The requirement that the sampling distribution of $\bar{x}$ is approximately normal if either the population variable $X$ is normally distributed or the sample size is $n \ge 30$.

Sampling Distribution of the Proportion Normality Condition

The requirement that the sampling distribution of $\hat{p}$ is approximately normal only when both expected counts meet the threshold: $np \ge 5$ and $n(1 - p) \ge 5$.

Standard Error of the Mean ($\sigma_{\bar{x}}$)

The standard deviation of the sampling distribution of sample means, measuring sample spread as $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$, centered at $\mu_{\bar{x}} = \mu$.

Standard Error of the Proportion ($\sigma_{\hat{p}}$)

The standard deviation of the sampling distribution of sample proportions, measuring sample spread as $\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}$, centered at $\mu_{\hat{p}} = p$.

Z-Score (Sampling Distribution)

A standardized measure describing how many standard deviations a sample statistic is from its distribution center, calculated as $Z = \frac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}}$ for a mean or $Z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}}$ for a proportion.

Population Parameter

A fixed numerical characteristic that describes an entire population or originates from a prior study (such as $\mu$, $\sigma$, or $p$), rather than being computed from the current sample.

Sample Statistic

A numerical descriptive value calculated directly from the observations in the current sample (such as $\bar{x}$, $S$, or $\hat{p}$) used to estimate an unknown population parameter.

Confidence Interval

A range of reasonable values for an unknown population parameter calculated from sample data using the formula $\text{point estimate} \pm \text{margin of error}$.

Point Estimate

The single best numerical value obtained from sample data that serves as the center of a confidence interval ($\bar{x}$ for a mean, or $\hat{p}$ for a proportion).

Margin of Error ($e$)

The maximum expected distance above and below the point estimate in a confidence interval, calculated as the critical value multiplied by the standard error.

Degrees of Freedom ($df$)

The parameter used to determine critical values from the Student's $t$-distribution table when estimating a population mean with unknown $\sigma$, calculated as $df = n - 1$.

Confidence Interval for a Population Mean

The interval estimate $\bar{x} \pm e$, where $e = Z \times \frac{\sigma}{\sqrt{n}}$ when the population standard deviation $\sigma$ is known, or $e = t \times \frac{S}{\sqrt{n}}$ with $df = n - 1$ when $\sigma$ is unknown.

Confidence Interval for a Population Proportion

The interval estimate $\hat{p} \pm e$, where $e = Z \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, applicable when both sample condition checks satisfy $n\hat{p} \ge 5$ and $n(1 - \hat{p}) \ge 5$.

Sample Size Formula for a Mean

The formula used to determine the minimum sample size required to estimate a population mean for a chosen margin of error: $n = \frac{Z^2 \times \sigma^2}{e^2}$, where the result must always be rounded up to the next whole number.

Sample Size Formula for a Proportion

The formula used to determine the minimum sample size required to estimate a population proportion: $n = \frac{Z^2 \times p(1 - p)}{e^2}$, using $p = 0.5$ if no prior estimate is provided, and always rounding up to the next whole number.

Confidence Interval Width and Midpoint

Geometric properties of a confidence interval defined by limits $LL$ and $UL$, where $\text{Midpoint} = \frac{UL + LL}{2}$ and $\text{Width} = UL - LL = 2e$.

Null Hypothesis ($H_0$)

The baseline hypothesis representing the established belief or status quo in a hypothesis test that always contains an equality sign ($=$, $\le$, or $\ge$).

Alternative Hypothesis ($H_a$)

The research hypothesis representing the new claim or statement being investigated that never contains an equality sign ($\ne$, $>$, or $<$) and determines the direction of the test.

Type 1 Error ($\alpha$)

The error committed when the null hypothesis ($H_0$) is rejected when it is actually true (referred to as a 'false alarm').

Type 2 Error ($\beta$)

The error committed when the test fails to reject the null hypothesis ($H_0$) when it is actually false (referred to as a 'missed it').

Inverse Relationship of $\alpha$ and $\beta$

The foundational principle in hypothesis testing establishing that the probability of a Type 1 Error ($\alpha$) and the probability of a Type 2 Error ($\beta$) are inversely proportional.

Two-Tailed Test

A hypothesis test specified by $H_a: \mu \ne \mu_0$ with critical values $\pm Z_{\alpha/2}$ or $\pm t_{\alpha/2}$, where $H_0$ is rejected if the absolute value of the test statistic exceeds the critical value ($|TS| > CV$).

Right-Tailed Test

A hypothesis test specified by $H_a: \mu > \mu_0$ with critical value $Z_{\alpha}$ or $t_{\alpha}$, where $H_0$ is rejected if the test statistic exceeds the critical value ($TS > CV$).

Left-Tailed Test

A hypothesis test specified by $H_a: \mu < \mu_0$ with critical value $-Z_{\alpha}$ or $-t_{\alpha}$, where $H_0$ is rejected if the test statistic is less than the critical value ($TS < CV$).

Test Statistic for a Proportion ($Z$)

The standardized test statistic used for testing a population proportion: $Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$, requiring both $np_0 \ge 5$ and $n(1 - p_0) \ge 5$.

Hypothesis Testing Decision Rule

The formal criterion determining that if the calculated test statistic falls within the designated rejection region, the final decision is to reject $H_0$; otherwise, the decision is to do not reject $H_0$.

Hypothesis Testing Decision Flowchart

A structured 7-step process for hypothesis testing: 1. Identify parameter ($\mu$ or $p$); 2. Check normality conditions; 3. Choose test statistic ($Z$ vs. $t$); 4. State $H_0$ and $H_a$; 5. Identify test tails; 6. Find critical values; 7. State final rejection decision.