Paired t-test and Goodness-of-Fit

Flashcards covering key definitions, formulas, hypotheses, and assumptions for the paired t-test and Chi-Squared Goodness-of-Fit test from Week 11.

Observations are paired when each value in one sample is __________ to a value in the other sample.

linked

A common mistake when analyzing paired data is treating the paired data as __________ samples.

independent

In a paired $t$-test, the null hypothesis is specified as $H_0: \_\_\_\_\_\_\_\_\_\_$.

$$\mu_d = 0$$

For the paired $t$-test alternative hypothesis $H_1: \mu_d > 0$, the Excel function used to compute the $p$-value is __________.

T.DIST.RT()

The test statistic for a paired $t$-test, $t = \frac{\bar{d} - \mu_d}{s_d / \sqrt{n}}$, follows a $t$-distribution with __________ degrees of freedom.

$$n - 1$$

The paired $t$-test is fundamentally concerned with the __________, not whether every individual improves.

average difference

When calculating a paired $t$ confidence interval for $\mu_d$ with small sample sizes ($n$), we assume that the distribution of the differences is approximately __________.

normal

In a Goodness-of-Fit test, observed frequencies ($O$) are defined as the __________ recorded in each category.

actual number of observations

The formula for the Chi-Squared Goodness-of-Fit test statistic is $D^2 = \sum_{i=1}^k \_\_\_\_\_\_\_\_\_\_$.

$$\frac{(O_i - E_i)^2}{E_i}$$

The Chi-Squared Goodness-of-Fit statistic $D^2$ follows a $\chi^2$ distribution with __________ degrees of freedom, where $k$ is the number of categories and $d$ is the number of parameters estimated from the data.

$$k - d - 1$$

In a Goodness-of-Fit test, a small test statistic $D^2$ indicates that the sample data are a __________ to what we would expect if $H_0$ were true.

good fit

The decision rule for a Goodness-of-Fit test states that we reject $H_0$ if $D^2 > \chi^2_{\alpha,df}$ or if $p\text{-value} < \_\_\_\_\_\_\_\_\_\_$.

$$\alpha$$

A critical assumption of the Goodness-of-Fit test is that each expected frequency must satisfy $E_i \ge \_\_\_\_\_\_\_\_\_\_$.

$$5$$

While the paired $t$-test is designed for numerical data, the Goodness-of-Fit test is applied to __________ data.

categorical