Statistics: Hypothesis Testing with One and Two Samples - 11/2 - 11/4

Vocabulary flashcards covering one-sample and two-sample hypothesis testing concepts, decision rules, errors, test statistics, and calculator procedures.

P-Value

The probability that, if the null hypothesis is true, the results from another randomly selected sample will be as extreme as, or more extreme than, the results obtained from the given sample.

Significance Level ($\alpha$)

A preset value (often $0.05$, or $0.01$ for a $1\%$ level) chosen before collecting sample data to determine the threshold for rejecting the null hypothesis.

"If the P is low, the null must go"

A memory aid summarizing the decision rule that if the $P\text{-value}$ is lower than the significance level $\alpha$, the null hypothesis $H_0$ must be rejected.

"Do Not Reject the Null Hypothesis"

The conclusion reached when the $P\text{-value}$ is greater than or equal to $\alpha$, indicating that the sample data did not provide conclusive evidence to overturn the null hypothesis, rather than proving it is true.

Left-Tailed Test

A hypothesis test whose alternative hypothesis contains a "less than" symbol (such as $H_a: \mu < \mu_0$ or $H_a: p < p_0$), placing extreme values to the left of the center.

Right-Tailed Test

A hypothesis test whose alternative hypothesis contains a "greater than" symbol (such as $H_a: \mu > \mu_0$ or $H_a: p > p_0$), placing extreme values to the right of the center.

Two-Tailed Test

A hypothesis test whose alternative hypothesis contains a "not equal to" symbol ($\ne$), placing extreme values in both the left and right tails of the distribution.

Type I Error

The error committed when a true null hypothesis is rejected, such as concluding that Jeffrey swims faster with new goggles when his true mean time is actually $16.43\,\text{seconds}$.

Type II Error

The error committed when a false null hypothesis is not rejected, such as concluding that Jeffrey's new goggles did not help when they actually did.

Z-Test (Single Population Mean)

A hypothesis test conducted on the normal distribution for a single population mean when the population standard deviation ($\sigma$) is known and the population is normally distributed or $n > 30$.

Student's t-Distribution (Single Population Mean)

A continuous probability distribution used to test claims about a single population mean when the population standard deviation ($\sigma$) is unknown.

Degrees of Freedom ($df$ for Single Mean t-Test)

The parameter of the Student's t-distribution for a single population mean test, calculated using the formula $df = n - 1$, where $n$ is the sample size.

Sampling Distribution of the Sample Proportion ($P'$)

The normal distribution modeling sample proportions under the null hypothesis, expressed as $P' \sim N\left(p, \sqrt{\frac{pq}{n}}\right)$, where $p$ is the hypothesized proportion, $q = 1 - p$, and $n$ is the sample size.

1-PropZTest

A TI-83/84 calculator function located under STAT TESTS used to perform a hypothesis test for a single population proportion.

Test Statistic

The standardized $z\text{-score}$ or $t\text{-score}$ of a sample estimate relative to the hypothesized parameter value, centered at a mean of $0$.

Test Statistic Formula for Mean with Known $\sigma$

The formula used to compute the standard normal test statistic: $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, and $n$ is the sample size.

Test Statistic Formula for Mean with Unknown $\sigma$

The formula used to compute the Student's t test statistic: $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$, where $s$ represents the sample standard deviation.

Conditions for Normal Approximation of Proportions

The requirements that $np \ge 5$ and $nq \ge 5$ (with $q = 1 - p$), which must be satisfied to compute the proportion test statistic $z = \frac{p' - p}{\sqrt{\frac{pq}{n}}}$.

Independent Groups

Two or more groups where the sample values selected from one group are not related in any way to the sample values selected from the other group.

Dependent Groups

Matched or paired samples in which testing involves calculating differences and testing a single population mean of those differences.

Aspin-Welch t-Test

A hypothesis test used to compare two population means when the population standard deviations ($\sigma_1$ and $\sigma_2$) of two independent groups are unknown.

Difference of Two Sample Means ($\bar{X}_1 - \bar{X}_2$)

The primary random variable analyzed when testing hypotheses comparing two independent population means.

Standard Error for Two Independent Means

The estimated standard deviation of $\bar{X}_1 - \bar{X}_2$ when population standard deviations are unknown, calculated as $\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$.

Test Statistic Formula for Two Population Means (Unknown $\sigma$)

The $t\text{-score}$ formula used in the Aspin-Welch t-test: $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$.

2-SampTTest Pooled Option

A setting on the TI-83/84 calculator under 2-SampTTest that must be set to "No" when population standard deviations are unknown and not assumed to be equal.