BMMA120 Unit 6: Sampling Distributions and Estimation

Vocabulary flashcards covering sampling distributions, point and interval estimation, sample size determination, pooled variance formulas, and confidence intervals for means, proportions, and variance ratios.

Sampling Distribution

The probability distribution of a sample statistic (such as a sample mean or sample variance) that describes a numerical characteristic of sample data derived from a random sample.

Sampling Distribution of the Sample Mean ($\bar{x}$)

The distribution of sample means drawn from a normal population with mean $\mu$ and variance $\sigma^2$, having expectation $E(\bar{x}) = \mu$ and variance $\text{Var}(\bar{x}) = \frac{\sigma^2}{n}$, denoted as $\bar{x} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$.

Sampling Distribution of the Difference of Two Means

The distribution formed from two independent normal samples, expressed as $\bar{x}_1 - \bar{x}_2 \sim N\left(\mu_1 - \mu_2, \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}\right)$.

Variance Sum Law

The statistical result that dictates adding rather than subtracting sample variances when establishing the variance of the difference between two independent sample means.

Point Estimation

The statistical process of inferring the value of a population parameter using a single sample value (such as using $\bar{x}$ to estimate $\mu$), which does not communicate the uncertainty of the estimate.

Interval Estimation

The process of providing a range of values, known as a confidence interval, within which a population parameter is expected to fall alongside a specified level of confidence.

Confidence Interval (CI)

A range of values derived from a sample that is likely to contain an unknown population parameter, structured generally as $\text{Sample Statistic} \pm \text{Margin of Error}$.

Margin of Error (ME)

The component added to and subtracted from a sample statistic in a confidence interval to account for sampling variability.

Confidence Interval Width

The mathematical difference between the upper limit and the lower limit of a confidence interval, calculated as $2 \times \text{ME}$.

Sample Proportion ($\hat{p}$)

A sample statistic calculated as $\hat{p} = \frac{x}{n}$, where $x$ is the number of successes and $n$ is the total number of trials in a binomial experiment.

Large Sample CI for a Population Proportion

The confidence interval formula for a single proportion when $n$ is large: $\hat{p} \pm Z_{\frac{\alpha}{2}}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.

Minimum Sample Size for a Proportion ($n$)

The formula used to determine the minimum number of observations needed to achieve a specified margin of error $\text{ME}$ at a selected confidence level: $n = \frac{\left(Z_{\frac{\alpha}{2}}\right)^2 \hat{p}(1 - \hat{p})}{(\text{ME})^2}$.

Confidence Interval for a Single Mean (Small Sample, $\sigma$ Unknown)

The confidence interval used when $n < 30$ and the population standard deviation is unknown, defined as $\bar{x} \pm t_{\frac{\alpha}{2}, n-1}\cdot\frac{s}{\sqrt{n}}$, utilizing a $t$-distribution with $n - 1$ degrees of freedom.

Confidence Interval for a Single Mean (Small Sample, $\sigma$ Known)

The confidence interval formula $\bar{x} \pm Z_{\frac{\alpha}{2}}\cdot\frac{\sigma}{\sqrt{n}}$, where knowing the population parameter outweighs the small sample size condition ($n < 30$), allowing the use of standard normal $Z$-critical values.

Pooled Sample Variance ($S_p^2$)

A combined variance estimate used when two independent small samples are assumed to have equal population variances, calculated as $S_p^2 = \frac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2}$.

Simplified Pooled Sample Variance

The streamlined form of pooled variance when two independent samples have equal sizes ($n_1 = n_2 = n$), given by $S_p^2 = \frac{S_1^2 + S_2^2}{2}$.

Fee-Weighted Pooled Variance

An adapted pooled variance formula where sample variances are scaled by monetary or procedural weights ($W_1$ and $W_2$), computed for equal sample sizes as $S_p^2 = \frac{(W_1 S_1)^2 + (W_2 S_2)^2}{2}$.

Confidence Interval for the Ratio of Two Population Variances

An asymmetric confidence interval based on the non-symmetric $F$-distribution, calculated as $\left(\frac{F}{F_{\frac{\alpha}{2}, n_1-1, n_2-1}}, \frac{F}{1/F_{\frac{\alpha}{2}, n_1-1, n_2-1}}\right)$, where $F = \frac{s_1^2}{s_2^2}$ with $s_1^2 > s_2^2$.

Inclusion of Zero in Difference of Means CI

A statistical outcome where a confidence interval for $(\mu_1 - \mu_2)$ spans both negative and positive numbers, implying insufficient evidence of a statistically significant difference between the two treatments or groups.

Inclusion of One in Variance Ratio CI

A statistical outcome where the confidence interval for the ratio of two population variances contains the value $1$, indicating that the population variances can reasonably be assumed to be equal.