Chapter 5: Inferences Based on a Single Sample - Estimation with Confidence Intervals

Vocabulary flashcards covering core definitions, parameters, conditions, distributions, and sample-size formulas from Chapter 5 of 'A First Course in Statistics' (Twelfth Edition) by McClave and Sincich.

Target Parameter

The unknown population parameter (such as the mean $\mu$, proportion $p$, or variance $\sigma^2$) that we are interested in estimating.

Point Estimator

A rule or formula that tells us how to use sample data to calculate a single number that can be used as an estimate of the target parameter.

Interval Estimator (Confidence Interval)

A formula that tells us how to use sample data to calculate an interval that estimates the target parameter.

Confidence Coefficient

The probability that an interval estimator encloses the population parameter—that is, the relative frequency with which the interval estimator encloses the population parameter when the estimator is used repeatedly a very large number of times.

Confidence Level

The confidence coefficient expressed as a percentage, denoted as $100(1 - \alpha)\%$.

$$z_{\alpha}$$

The value of the standard normal random variable $z$ such that the area $\alpha$ lies to its right, satisfying $P(z > z_{\alpha}) = \alpha$.

Large-Sample Confidence Interval for $\mu$

An interval estimate for a population mean calculated when $n \ge 30$, defined as $\bar{x} \pm (z_{\alpha/2})\left(\frac{\sigma}{\sqrt{n}}\right)$ when $\sigma$ is known, or $\bar{x} \pm (z_{\alpha/2})\left(\frac{s}{\sqrt{n}}\right)$ when $\sigma$ is unknown.

Conditions for a Valid Large-Sample Confidence Interval for $\mu$

1. A random sample is selected from the target population. 2. The sample size $n$ is large ($n \ge 30$), guaranteeing by the Central Limit Theorem that the sampling distribution of $\bar{x}$ is approximately normal and $s$ is a good estimator of $\sigma$.

Student's $t$-Distribution

A bell-shaped, symmetric probability distribution centered at $0$ with heavier tails and greater variability than the standard normal distribution, parameterized by degrees of freedom ($df$) and approaching the $z$-distribution as $df$ approaches infinity.

Small-Sample Confidence Interval for $\mu$

An interval estimate based on Student's $t$-statistic when $\sigma$ is unknown and the sample is small, computed as $\bar{x} \pm (t_{\alpha/2})\left(\frac{s}{\sqrt{n}}\right)$, where $t_{\alpha/2}$ is based on $(n - 1)$ degrees of freedom.

Conditions for a Valid Small-Sample Confidence Interval for $\mu$

1. A random sample is selected from the target population. 2. The population has a relative frequency distribution that is approximately normal.

Sampling Distribution of $\hat{p}$

The probability distribution of the sample proportion $\hat{p}$, which has a mean of $p$ (unbiased), a standard deviation of $\sigma_{\hat{p}} = \sqrt{\frac{pq}{n}}$ where $q = 1 - p$, and is approximately normal when both $n\hat{p} \ge 15$ and $n\hat{q} \ge 15$.

Large-Sample Confidence Interval for $p$

An interval estimate for a population proportion calculated as $\hat{p} \pm z_{\alpha/2}\sqrt{\frac{\hat{p}\hat{q}}{n}}$, where $\hat{p} = \frac{x}{n}$ and $\hat{q} = 1 - \hat{p}$.

Conditions for a Valid Large-Sample Confidence Interval for $p$

1. A random sample is selected from the target population. 2. The sample size $n$ is large, satisfied when both the number of successes ($n\hat{p}$) and failures ($n\hat{q}$) are at least $15$.

Adjusted Confidence Interval for a Population Proportion $p$

A confidence interval given by $\tilde{p} \pm z_{\alpha/2}\sqrt{\frac{\tilde{p}(1 - \tilde{p})}{n + 4}}$, where $\tilde{p} = \frac{x + 2}{n + 4}$ is the adjusted sample proportion of successes.

Sampling Error ($SE$)

The half-width of a confidence interval representing the maximum tolerable bound on the error of estimation at a specified confidence level.

Sample Size for Estimating $\mu$

The required number of observations to estimate $\mu$ within a specified sampling error $SE$ at $100(1 - \alpha)\%$ confidence: $n = \frac{(z_{\alpha/2})^2\sigma^2}{(SE)^2}$, rounding the result upward.

Sample Size for Estimating $p$

The required number of observations to estimate $p$ within a specified sampling error $SE$ at $100(1 - \alpha)\%$ confidence: $n = \frac{(z_{\alpha/2})^2(pq)}{(SE)^2}$, rounded upward (often using $p = 0.5$ for a conservative upper bound on $pq$).

Chi-Square Distribution ($\chi^2$)

A right-skewed, non-negative continuous probability distribution characterized by degrees of freedom, used in constructing confidence intervals for a population variance $\sigma^2$.

Confidence Interval for $\sigma^2$

A $100(1 - \alpha)\%$ interval estimate for a population variance given by $\frac{(n - 1)s^2}{\chi^2_{\alpha/2}} \le \sigma^2 \le \frac{(n - 1)s^2}{\chi^2_{(1 - \alpha/2)}}$, based on $(n - 1)$ degrees of freedom.

Conditions for a Valid Confidence Interval for $\sigma^2$

1. A random sample is selected from the target population. 2. The population of interest has a relative frequency distribution that is approximately normal.

Nonparametric Statistical Methods

Statistical procedures recommended when the assumption of population normality is severely violated for small samples.