Central Limit Theorem, Normal Approximation, and Confidence Intervals - 10/12 - 10/14

Vocabulary flashcards covering the Central Limit Theorem, the normal approximation to the binomial distribution with continuity corrections, and confidence intervals for a single population mean.

Central Limit Theorem Sample Size Condition

The condition stating that the Central Limit Theorem applies to the distribution of sample means if the sample size is greater than or equal to 30 ($n \ge 30$).

Uniform Distribution Mean Formula

The formula used to calculate the mean of a continuous uniform distribution on the interval $[a, b]$, given by

$\mu_X = \frac{a+b}{2}$.

Uniform Distribution Standard Deviation Formula

The formula used to compute the standard deviation of a continuous uniform distribution on the interval $[a, b]$, given by

$\sigma_X = \frac{b-a}{\sqrt{12}}$.

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

The probability distribution of the mean of random samples of size $n$, defined under the Central Limit Theorem as

$\bar{X} \sim N\left(\mu_X, \frac{\sigma_X}{\sqrt{n}}\right)$.

Sampling Distribution of the Sample Sum ($\sum X$)

The probability distribution of the total or sum of values from random samples of size $n$, defined under the Central Limit Theorem as $\sum X \sim N\left(n\mu_X, \sqrt{n}\sigma_X\right)$.

Interquartile Range ($IQR$)

The measure of statistical dispersion determined by calculating the difference between the third quartile and the first quartile: $IQR = Q_3 - Q_1$.

Normal Approximation to Binomial Distribution Condition

The criteria specifying that a discrete binomial variable $X \sim B(n, p)$ can be approximated by a normal distribution if both $np > 5$ and $nq > 5$, where $q = 1 - p$.

Approximating Mean for Binomial Distribution ($\mu$)

The mean parameter used when approximating a binomial distribution with a normal distribution, calculated as $\mu = np$.

Approximating Standard Deviation for Binomial Distribution ($\sigma$)

The standard deviation parameter used when approximating a binomial distribution with a normal distribution, calculated as $\sigma = \sqrt{npq}$.

Continuity Correction for "At Least" ($X \ge a$)

The continuity adjustment applied when approximating a discrete binomial probability with a normal distribution, rewriting $P(X \ge a)$ as the continuous probability $P(X > a - 0.5)$.

Continuity Correction for "At Most" ($X \le a$)

The continuity adjustment applied when approximating a discrete binomial probability with a normal distribution, rewriting $P(X \le a)$ as the continuous probability $P(X < a + 0.5)$.

Continuity Correction for "More Than" ($X > a$)

The continuity adjustment applied when approximating a discrete binomial probability with a normal distribution, rewriting $P(X > a)$ as the continuous probability $P(X > a + 0.5)$.

Continuity Correction for "Fewer Than" ($X < a$)

The continuity adjustment applied when approximating a discrete binomial probability with a normal distribution, rewriting $P(X < a)$ as the continuous probability $P(X < a - 0.5)$.

Continuity Correction for "Exactly" ($X = a$)

The continuity adjustment applied when approximating a discrete binomial probability with a normal distribution, rewriting $P(X = a)$ as the continuous interval $P(a - 0.5 < X < a + 0.5)$.

Point Estimate

A single numerical sample value, such as the sample mean $\bar{x}$, used as the best estimate of an unknown population parameter like $\mu$.

Standard Error of the Mean ($\sigma_{\bar{x}}$, or $s_{\bar{x}}$)

The standard deviation of the sampling distribution of the sample mean, calculated by dividing the population standard deviation by the square root of the sample size:

$\frac{\sigma}{\sqrt{n}}$ .

Empirical Rule for Sample Means

The principle stating that approximately 95% of sample means are within two standard deviations of the true population mean ($\bar{x} \pm 2\sigma_{\bar{x}}$).

Confidence Level ($CL$)

The stated proportion or percentage (such as 0.90, 0.95, or 0.99) representing the degree of certainty that a constructed confidence interval will contain the true population parameter.

Alpha ($\alpha$)

The total area in both tails outside the specified confidence level under a standard normal distribution curve, calculated as $\alpha = 1 - CL$.

Critical Value ($z_{\alpha/2}$)

The critical z-score boundary derived from standard normal distribution $N(0, 1)$ that cuts off an area of

$\frac{\alpha}{2}$ in the upper tail.

Error Bound for a Population Mean ($EBM$)

The margin of error representing the maximum expected distance between the sample mean and the true population mean, calculated as

$EBM = z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)$.

Confidence Interval for a Population Mean

An interval estimate constructed around the point estimate to capture the true population mean, expressed in the form $(\bar{x} - EBM, \bar{x} + EBM)$.

Critical Value for a 90% Confidence Level ($z_{0.05}$)

The critical z-score corresponding to a 90% confidence level ($\alpha = 0.10$, tail area $0.05$), calculated using $\text{invNorm}(0.95, 0, 1) \approx 1.6449$.

Critical Value for a 95% Confidence Level ($z_{0.025}$)

The critical z-score corresponding to a 95% confidence level ($\alpha = 0.05$, tail area $0.025$), calculated using $\text{invNorm}(0.975, 0, 1) \approx 1.9600$.

Cumulative Binomial Probability Function (binomcdf)

The graphing calculator command used to evaluate cumulative binomial probabilities $P(X \le k)$ given parameters $n$, $p$, and upper limit $k$.