Chapter 8: Confidence Intervals

A set of vocabulary flashcards reviewing Chapter 8: Confidence Intervals for a single population mean using the normal distribution.

Reverse-direction problem

A scenario where the population mean $\mu$ is unknown, and an estimate of $\mu$ is sought by selecting a sample of size $n$ and calculating $\overline{x}$.

Point estimate

The sample mean ($\overline{x}$), which serves as a single-value estimate for the unknown population mean ($\mu$).

Population mean ($\mu$)

The unknown population measure that a researcher aims to estimate using sample data.

Sample mean ($\overline{x}$)

The point estimate of the population mean $\mu$ obtained from a sample of size $n$, serving as the central value of a confidence interval.

Key Question of Confidence Intervals

The central question: 'How do we know how accurately $\overline{x}$ approximates $\mu$?'

Empirical Rule

A rule stating that in approximately $95\%$ of samples, the population mean $\mu$ is within two standard deviations of $\overline{x}$.

Empirical Rule interval limits

The boundaries $\overline{x} - 2s_{\overline{x}}$ and $\overline{x} + 2s_{\overline{x}}$, between which $\mu$ can be found in approximately $95\%$ of samples.

$$s_{\overline{x}}$$

The standard deviation of the sample mean, calculated as $s_{\overline{x}} = \frac{\sigma_x}{\sqrt{n}}$.

Distribution of $\overline{X}$

The sampling distribution of the sample mean, given as $\overline{X} \sim N\left(\mu_x, \frac{\sigma_x}{\sqrt{n}}\right)$.

"2" in the Empirical Rule

The number of standard deviations corresponding to the confidence level, representing the $z$-score.

Confidence Level (CL)

A given probability or percentage (such as $0.95$, $90\%$, $95\%$, or $99\%$) representing the degree of certainty for a confidence interval.

Error bound for the population mean (EBM)

The margin of error added to and subtracted from the point estimate, exemplified by the quantity $2s_{\overline{x}}$.

$$2s_{\overline{x}}$$

An example of the Error Bound for the population mean (EBM) derived from the Empirical Rule for approximately a $95\%$ confidence level.

Central Value

The sample mean ($\overline{x}$), which forms the center of the confidence interval.

Confidence interval form

The standard notation for reporting the final confidence interval answer: $(\overline{x} - \text{EBM}, \overline{x} + \text{EBM})$.

Typical Confidence Levels

The standard confidence levels provided in problems: $90\%$, $95\%$, and $99\%$.

Population standard deviation ($\sigma$)

A known population parameter provided in typical problems to calculate the standard error and the EBM.

Sample size ($n$)

The number of observations in a sample, which appears under the radical sign ($\sqrt{n}$) in the standard error formula.

$\alpha$ (Alpha)

The complement of the confidence level, calculated in Step 1 as $\alpha = 1 - \text{CL}$.

$$z_{\alpha/2}$$

The critical $z$-score associated with a specific confidence level, denoted $z_{\alpha/2}$.

Inverse norm feature (invNorm)

The graphing calculator command used in Step 2 to determine the critical $z$-score ($z_{\alpha/2}$).

EBM formula

The formula used to compute the error bound for the population mean: $\text{EBM} = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$.

Step 1 (Confidence Interval Procedure)

Find $\alpha$ using the equation $\alpha = 1 - \text{CL}$.

Step 2 (Confidence Interval Procedure)

Find the $z$-score (denoted $z_{\alpha/2}$) by using the Inverse norm feature in the graphing calculator.

Step 3 (Confidence Interval Procedure)

Compute the EBM using the formula $\text{EBM} = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$.