Business Statistics: Confidence Intervals and Hypothesis Testing

Vocabulary flashcards reviewing confidence intervals, one-sample hypothesis testing, critical values, and Excel statistical functions.

Sample Statistic

A numerical summary computed from a sample (such as a sample mean of $17.8\text{ minutes}$ from $180\text{ students}$) used to estimate an unknown population parameter.

Confidence Interval Setup (Known $\sigma$)

The mathematical formula $\bar{x} \pm z^*\left(\frac{\sigma}{\sqrt{n}}\right)$ used to construct an interval estimate for a population mean when the population standard deviation $\sigma$ is known.

95% Confidence Interval Interpretation

The operational interpretation that the statistical estimation method captures the fixed population mean in approximately $95\%$ of repeated random samples from the same population.

Excel Two-Sided Normal Critical Value

The Excel formula =NORM.S.INV(1-0.01/2) (or =NORM.S.INV(0.995)) used to obtain the positive critical value $z^*$ for a two-sided $99\%$ confidence interval.

Degrees of Freedom ($df$) for One-Mean $t$-Interval

The degrees of freedom parameter for Student's $t$-distribution calculated as $df = n - 1$ when estimating a mean with an unknown population standard deviation $\sigma$.

Count Check for Proportion Confidence Interval

The requirement that a sample contains at least $10$ observed successes ($n\hat{p} \ge 10$) and at least $10$ observed failures ($n(1-\hat{p}) \ge 10$) to justify normal approximation.

Sample Size Planning Formula (Proportion)

The formula $n = \left(\frac{z^*}{E}\right)^2 p^*(1-p^*)$, which determines the minimum whole sample size needed to achieve a specified margin of error $E$ with confidence coefficient $z^*$ and planning proportion $p^*$.

Alternative Hypothesis ($H_a$)

The claim about a population parameter that represents the research objective, such as $H_a: \mu < 30$ indicating a left-tailed test that the mean has fallen below $30$.

Type I Error

The mistake of rejecting the null hypothesis $H_0$ when it is actually true (for example, concluding a defect rate increased when the benchmark still holds).

One-Sample $z$-Statistic (Known $\sigma$)

The standardized test statistic calculated as $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$ to test a null hypothesis mean $\mu_0$ against an observed sample mean $\bar{x}$.

Excel Left-Tailed Normal $p$-Value

The formula =NORM.S.DIST(z, TRUE) (such as =NORM.S.DIST(-1.70, TRUE)) used to find the cumulative left-tail probability for an observed test statistic under the standard normal distribution.

$p$-Value Decision Rule

The decision criterion stating that if the $p$-value is less than or equal to the significance level $\alpha$ ($p \le \alpha$), the null hypothesis $H_0$ is rejected in favor of the alternative $H_a$.

Confidence Interval and Two-Sided Test Connection

The mathematical equivalence stating that a two-sided test rejects $H_0: \mu = \mu_0$ at significance level $\alpha$ if and only if the hypothesized value $\mu_0$ falls outside the corresponding $(1-\alpha)\times 100\%$ confidence interval.

One-Sample $t$-Statistic (Unknown $\sigma$)

The test statistic calculated as $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ with $df = n - 1$, used for hypothesis testing on a single mean when the population standard deviation is unknown.

Excel Right-Tailed $t$ $p$-Value

The Excel formula =T.DIST.RT(t, df) (such as =T.DIST.RT(1.85, 19)) used to calculate the right-tail probability for an observed $t$-statistic with specified degrees of freedom.

Expected Count Condition (Proportion Test)

The requirement in a one-proportion $z$-test to verify that expected successes $n p_0$ and expected failures $n(1-p_0)$ under the null hypothesis are both at least $10$.

Standard Error in Proportion Testing vs. Estimation

The methodological rule requiring the null proportion $p_0$ inside the standard error $\sqrt{\frac{p_0(1-p_0)}{n}}$ for hypothesis tests, but the sample proportion $\hat{p}$ inside $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ for confidence intervals.

One-Proportion $z$-Statistic

The standardized test statistic computed as $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ to evaluate whether a sample proportion $\hat{p}$ differs significantly from a hypothesized null proportion $p_0$.

Excel Right-Tailed Normal $p$-Value

The Excel expression =1-NORM.S.DIST(z, TRUE) used to determine the upper-tail probability for a right-tailed one-proportion $z$-test with an observed positive $z$-value.

Excel Left-Tailed $t$ $p$-Value

The Excel function =T.DIST(t, df, TRUE) (such as =T.DIST(-1.40, 19, TRUE)) used to compute the cumulative left-tail area for a negative observed $t$-statistic.