Detailed STAT 503 - Lesson 2: Simple Comparative Experiments Practice Flashcards

A comprehensive vocabulary-style flashcard set covering Lesson 2 of STAT 503, including t-tests, sample size determination, power analysis, and paired designs.

Lesson 2 purpose

To review the basic assumptions and foundations of estimation and hypothesis testing before moving into advanced Design of Experiments topics.

Lesson objective 1

Review basic statistical concepts.

Lesson objective 2

Review sample-size calculations for two-sample problems based on the t-test.

Lesson objective 3

Review the difference between two independent samples and a paired-comparison design.

Lesson objective 4

Review assumptions underlying the t-test and how those assumptions can be assessed.

Simple comparative experiment

An experiment comparing two conditions or groups via hypothesis testing, estimation, and sample-size determination.

Two-sample experiment example

A study comparing two formulations for making cement mortar.

Reason for graphing data

Graphical displays reveal distributional shape, spread, location, and possible outliers that tables of numbers alone hide.

Dot plot

A graphical display useful for seeing the distribution of observations, especially for very small data sets.

Boxplot

A graphical tool useful for both small and larger data sets that provides a quick snapshot of the distribution.

Box in a boxplot

Spans the middle $50\%$ of the data, from the $25$th percentile ($Q_1$) to the $75$th percentile ($Q_3$).

Median in a boxplot

The horizontal line inside the box.

Interquartile range (IQR)

$IQR = Q_3 - Q_1$, representing the width covered by the middle $50\%$ of observations.

Boxplot whiskers

Extend toward the minimum and maximum constants based on the usual $1.5 \times IQR$ rule.

Potential boxplot outliers

Individual observations plotted as dots beyond the whiskers.

Normal-data boxplot expectation

For reasonably normal data, one expects primarily the box and whiskers without isolated points beyond them.

Visual evidence versus statistical significance

Visualization can suggest differences, but does not alone establish a statistically significant difference between population means.

Cement example visual finding

The two groups appear different in median, and the dot plot/boxplots hint at a difference between formulations.

Two-sample t-test population assumption

Both samples are assumed to arise from Normal populations.

Two-sample population means notation

$\mu_1$ and $\mu_2$.

Equal-variance assumption

For the pooled two-sample t procedure, the population variances are assumed equal.

Unequal variances strategy

Transform the data to a scale with equal variation as more complex ANOVA models also assume equal variances.

Unequal-variance t-test caveat

While a t-test version handles unequal variances, it does not extend in the same way to complex ANOVA models.

Sample mean formula

$$\bar{y} = \frac{\sum_{i=1}^{n} y_i}{n}$$

Sample variance formula

$$s^2 = \frac{\sum_{i=1}^{n} (y_i - \bar{y})^2}{n - 1}$$

Purpose of dividing by $n - 1$

It makes $s^2$ an unbiased estimator of the population variance $\sigma^2$.

Three sufficient two-sample summaries

Sample size $n$, sample mean, and sample standard deviation (or variance).

Danger of summaries alone

They do not reveal distribution shape or potential outliers.

Two-sample t-test intuition

Compare the difference between sample means with the standard error of that difference.

Signal-to-noise interpretation

The t statistic is the difference between groups relative to within-group variation.

Why use t rather than z

Sample estimates replace unknown true population variances.

Student t distribution origin

Published by William Sealy Gosset under the pseudonym 'Student'.

Pooled variance formula ($s_p^2$)

$$s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$$

Pooled standard deviation formula ($s_p$)

$$s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_1^2}{n_1 + n_2 - 2}}$$

Standard error of two independent means

$$SE(\bar{y}_1 - \bar{y}_2) = s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$$

Two-sample null hypothesis

$H_0: \mu_1 = \mu_2$ or $H_0: \mu_1 - \mu_2 = 0$.

Two-sided alternative hypothesis

$H_1: \mu_1 \neq \mu_2$ or $H_1: \mu_1 - \mu_2 \neq 0$.

Two-sample t statistic formula

$$t = \frac{\bar{y}_1 - \bar{y}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$

Degrees of freedom for pooled t-test

$$df = n_1 + n_2 - 2$$

Magnitude of t statistic

Measures how far apart sample means are relative to within-group variation.

p-value

Under $H_0$, the probability of observing a test statistic at least as extreme as the one obtained.

Cement example t statistic

$$t = -2.19$$

Cement example p-value

$$p = 0.043$$

Cement example degrees of freedom

$$df = 17$$

Cement treatment 1 summary statistics

$n = 10$, mean = $16.764$, standard deviation = $0.316$, SE mean = $0.10$.

Cement treatment 2 summary statistics

$n = 10$, mean = $17.042$, standard deviation = $0.248$, SE mean = $0.078$.

Estimated mean difference

Treatment 1 - Treatment 2 = $-0.278$ in the cement example.

95% confidence interval in cement example

$(-0.546, -0.010)$ for $\mu_1 - \mu_2$.

Normal probability plot finding (Cement)

The lesson states that the normal probability plots look reasonable.

Confidence interval

An interval constructed to contain the true parameter with a specified confidence level (e.g., $95\%$).

Two-sample pooled confidence interval formula

$$(\bar{y}_1 - \bar{y}_2) \pm t_{1-\alpha/2, df} \times s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_1}}$$

Two-sided test and CI connection

Rejecting $H_0: \mu_1 - \mu_2 = 0$ corresponds to a confidence interval for the difference that does not contain $0$.

Cement CI interpretation

The $95\%$ CI excludes $0$, agreeing with the rejection of the null hypothesis at $\alpha = 0.05$.

Bound on the margin of error (B)

The desired accuracy for the estimate of the difference in population means.

Equal sample-size assumption for planning

The derivation assumes $n$ observations in each of the two groups.

Equal-n CI formula

$$(\bar{y}_1 - \bar{y}_2) \pm t_{1-\alpha/2, df} \times s \times \sqrt{\frac{2}{n}}$$

Equal-n degrees of freedom

$$df = n + n - 2 = 2(n - 1)$$

Margin-of-error equation (B)

$$B = t_{1-\alpha/2, df} \times s \times \sqrt{\frac{2}{n}}$$

Sample-size equation from B

$$n = \left[ \frac{t_{1-\alpha/2, df} \times s \times \sqrt{2}}{B} \right]^2$$

Unknown variance planning requirement

An estimate or guesstimate of $\sigma$ or $\sigma^2$ must be used prior to data collection.

Why z is used for hand calculation

If $n$ is unknown, $df$ and the exact $t$ critical value are unknown; $z$ provides an initial substitute.

Information needed for sample size

An estimate of $\sigma^2$, a bound $B$, and confidence level $1 - \alpha$.

Binomial worst-case variance

$p(1-p)$ is maximized at $p = 0.5$.

Binomial approximate sample-size formula

For $\alpha = 0.05$, the lesson gives $n = \frac{2}{B^2}$ for a two-group extension.

Paired samples

Two measurements taken on each subject or matched unit so that observations within a pair are correlated.

Cost of ignoring pairing

Subject-to-subject variability is treated as error, losing the efficiency of the matching.

Paired t-test analysis principle

Analyze the within-subject differences rather than treating the sets as independent.

Blocking in paired t-tests

A technique where the subject/patient acts as a block to remove nuisance variation.

Power

The probability of rejecting the null hypothesis when the alternative hypothesis is true ($1 - \beta$).

Type I error

Rejecting $H_0$ when $H_0$ is actually true, with probability $\alpha$.

Type II error

Failing to reject $H_0$ when $H_A$ is actually true, with probability $\beta$.

Operating characteristic (OC) curves

Curves used traditionally to determine $\beta$ and power for the t-test.

Standardized mean difference

$\frac{|\mu_1 - \mu_2|}{\sigma}$, used as the signal-to-noise ratio for OC curves.

OC-curve n-star formula

For the lesson's $\alpha = 0.05$ curve, $n^* = 2n - 1$.

Minitab power menu path

Stat > Power and Sample Size > 2-sample t.