Confidence Intervals for a Single Population Mean Using the Normal Distribution - 10/19

Vocabulary practice flashcards covering Section 8.1 on confidence intervals for a single population mean, critical values, sample size determination, and error bounds.

Confidence Level ($\text{CL}$)

The process success rate representing the proportion of times that the confidence intervals produced by a given procedure will, in the long run, contain the true population parameter (e.g., $90\%$, $95\%$, or $99\%$).

Error Bound for the Population Mean ($\text{EBM}$)

The margin of error added to and subtracted from the sample mean to construct a confidence interval, calculated using the formula $EBM = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.

Confidence Interval Form

The standard interval representation for a population mean written as $(\bar{x} - EBM, \bar{x} + EBM)$, where $\bar{x}$ is the sample mean and $EBM$ is the error bound for the mean.

Alpha ($\alpha$)

The total area under the standard normal curve that falls outside the confidence level, determined by the formula $\alpha = 1 - \text{CL}$ and split equally into two tails of size $\frac{\alpha}{2}$.

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

A fixed $z$-score associated with a specified confidence level, calculated on a graphing calculator using the inverse norm function $\text{invNorm}\left(1 - \frac{\alpha}{2}, 0, 1\right)$ or $\text{invNorm}\left(\frac{\alpha}{2} + \text{CL}, 0, 1\right)$.

Critical Value for $90\%$ Confidence Level

The critical value corresponding to $\alpha = 0.10$, calculated as $\text{invNorm}(0.95, 0, 1) = 1.644853626\dots$ and rounded to $z_{\alpha/2} \approx 1.6449$.

Critical Value for $95\%$ Confidence Level

The critical value corresponding to $\alpha = 0.05$, calculated as $\text{invNorm}(0.975, 0, 1) = 1.959963986\dots$ and rounded to $z_{\alpha/2} \approx 1.9600$.

Critical Value for $99\%$ Confidence Level

The critical value corresponding to $\alpha = 0.01$, calculated as $\text{invNorm}(0.995, 0, 1) = 2.575829303\dots$ and rounded to $z_{\alpha/2} \approx 2.5758$.

Correct Interpretation of a Confidence Interval

The formal statement that we are $\text{CL}\%$ confident that the computed interval actually does contain the true value of the population mean, reflecting confidence in the process rather than a probability of the fixed mean changing.

Population Mean ($\mu$)

A fixed, constant value that describes a characteristic of the entire population; it is not a random variable and does not move around.

Effect of Confidence Level on $\text{EBM}$

A direct relationship in which an increase in the confidence level ($\text{CL}$) causes an increase in the error bound for the mean ($\text{EBM}$), resulting in a wider interval.

Effect of Sample Size on $\text{EBM}$

An inverse relationship in which an increase in the sample size ($n$) causes a decrease in the error bound for the mean ($\text{EBM}$), resulting in a narrower interval.

Sample Mean from a Confidence Interval (Working Backwards)

The formula used to find the sample mean $\bar{x}$ given the lower and upper limits of a confidence interval: $\bar{x} = \frac{\text{lower limit} + \text{upper limit}}{2}$.

Sample Size Formula for Estimating a Population Mean

The formula $n = \left(\frac{z_{\alpha/2} \cdot \sigma}{EBM}\right)^2$, derived by solving the error bound equation for $n$, where the resulting value is always rounded up to the next integer.

Specific Absorption Rate (SAR)

A measure of the amount of radio frequency (RF) energy absorbed by the user's body when using a cell phone handset, which must be no more than $1.6\,\text{watts per kilogram}$ to receive FCC certification in the United States.