Stats Recitation: Sampling Distributions, Confidence Intervals, and t-Intervals

Vocabulary and formula flashcards covering Central Limit Theorem conditions, sampling distribution transformations, z-intervals, sample size formulas, and t-intervals based on recitation notes.

Central Limit Theorem condition

The sample size requirement where $n \ge 30$.

Standard error of the mean ($\sigma_{\bar{x}}$) formula

The standard deviation of the sampling distribution when sample size $n$ is introduced, calculated as $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.

Sampling distribution notation shift

When sample size $n$ is introduced to a problem, the mean $m$ and standard deviation $\sigma$ are replaced by $m_{\bar{x}}$ and $\sigma_{\bar{x}}$.

Alpha ($\alpha$)

The probability of error outside the confidence level, calculated as $\alpha = 1 - \text{Confidence Level}$.

$z$-confidence interval formula

$$\bar{x} \pm z_{\alpha/2}\left[\frac{\sigma}{\sqrt{n}}\right]$$

Choice between $z$ and $t$ distributions

Use $z$ when the population standard deviation ($\sigma$) is known; use $t$ when $n$ is small and sample standard deviation ($s$) is used.

Relationship between confidence level and interval length

As the confidence level increases (for example, from $90\%$ to $95\%$), the confidence interval becomes longer.

Sample size determination formula for a bound ($B$)

$$n = \left(\frac{\sigma z_{\alpha/2}}{B}\right)^2$$

Degrees of freedom ($df$) for a single-sample $t$-interval

Calculated as $df = n - 1$.

$t$-confidence interval formula

$$\bar{x} \pm t_{\alpha/2}\left[\frac{s}{\sqrt{n}}\right]$$