stat

Sampling design primary question

How did the individuals get into the study?

Experimental design primary question

How were treatments assigned to the experimental units?

Voluntary response sample

A sampling method where individuals select themselves to participate by responding to an open invitation.

Simple random sample (SRS)

A sampling method in which every possible set of $n$ individuals in the population has an equal chance of being selected.

Stratified random sampling

A sampling method in which the population is divided into distinct groups and a random sample is taken from every group.

Multistage sampling

A design in which groups are selected at random first, and then individuals are selected at random from within the chosen groups.

Completely randomized design

An experimental setup where all experimental units are allocated across treatments purely by chance without any prior grouping.

Randomized block design

An experimental setup where units are first grouped by a variable expected to influence the response, and then randomized to treatments within each group.

Matched pairs design

A specific blocking design featuring blocks of size two, or where each subject receives both treatments in random order.

Key distinction between stratified sampling and blocking

The former uses groups to select which subjects enter the study, whereas the latter uses groups to allocate which treatments subjects receive.

Experimental unit

The smallest individual entity that receives an experimental treatment.

Factor

An explanatory variable manipulated or set by the experimenter.

Treatment

A specific experimental condition applied to units, formed by a particular combination of factor levels.

Confounding

A condition occurring when the effect of an experimental treatment cannot be separated from the effect of an outside variable.

Calculation of Q1 and Q3 with an odd sample size

The medians of the lower and upper halves of the ordered data, strictly excluding the overall median from both halves.

Fences for identifying outliers via IQR

Lower fence at $Q_1 - 1.5 \times IQR$ and upper fence at $Q_3 + 1.5 \times IQR$.

Resistant summary measures

Median, quartiles, and $IQR$, because their values are unaffected by extreme observations.

Non-resistant summary measures

Mean, range, standard deviation, and variance.

Distribution skewness indicated by mean > median

Right-skewed.

Stemplot (Stem-and-leaf display)

A quantitative data display ideal for small datasets that illustrates shape while retaining every individual numerical value.

Boxplot

A compact graphical display of the five-number summary that highlights identified outliers.

Histogram

A graphical display for large quantitative datasets that shows overall distribution shape while individual data values are lost.

Standard error of the sample mean

$\frac{\sigma}{\sqrt{n}}$, which decreases as the sample size $n$ increases.

Central Limit Theorem (CLT)

As sample size $n$ becomes sufficiently large ($n \ge 30$), the sampling distribution of the sample mean becomes approximately Normal regardless of the population distribution shape.

68-95-99.7 rule

In a Normal distribution, approximately $68\%$ of observations lie within $\mu \pm \sigma$, $95\%$ within $\mu \pm 2\sigma$, and $99.7\%$ within $\mu \pm 3\sigma$.

Standardized score (z-score) formula

$$z = \frac{x - \mu}{\sigma}$$

One-sample z-test statistic for a population mean

$$z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$$

P-value decision rule

Reject $H_0$ if the $p$-value is less than or equal to $\alpha$; otherwise, fail to reject $H_0$.

Two-sided confidence interval matching a one-sided test at alpha = 0.05

A two-sided $90\%$ confidence interval.