Statistics C1000: Confidence Intervals and One-Sample Hypothesis Testing - 10/26 - 10/28

Comprehensive vocabulary flashcards covering Student's t-distributions, confidence intervals for population means and proportions, sample size calculation, and foundational concepts of one-sample hypothesis testing.

Student's t-distribution

A continuous probability distribution discovered by William Gosset at Guinness Brewery that has a mean of 0, is symmetric about the mean, and forms a bell-shaped curve with thicker tails and a lower center than the standard normal distribution.

Degrees of Freedom ($df$)

A parameter calculated as $df = n - 1$ (sometimes denoted $\nu$) that defines the shape of a Student's t-distribution; as this value increases, the t-distribution approaches the shape of the standard normal distribution.

William Gosset

The statistician working at Guinness Brewery who discovered the Student's t-distribution.

invT

A TI-84 calculator command written as invT(area to the left, degree of freedom) located directly below the inverse norm command, used to compute the critical value $t_{\alpha/2}$.

Error Bound for a Population Mean ($EBM$)

The margin of error when estimating a population mean with unknown population standard deviation, calculated using the formula $EBM = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$.

Sample Proportion ($P'$)

A statistic representing the fraction of successes in a sample, computed as $P' = \frac{X}{n}$ where $X$ is the count of successes and $n$ is the sample size; it is also denoted by $\hat{p}$.

Normal Approximation to the Binomial Distribution

An approximation applied when $np > 5$ and $nq > 5$ (where $q = 1 - p$), allowing a binomial variable $X \sim B(n, p)$ to be treated as normally distributed with mean $\mu = np$ and standard deviation $\sigma = \sqrt{npq}$.

Distribution of the Sample Proportion

The sampling distribution of $P'$, which is normally distributed as $P' \sim N\left(p, \sqrt{\frac{pq}{n}}\right)$ with a mean of $p$ and a standard deviation of $\sqrt{\frac{pq}{n}}$.

Error Bound for the Proportion ($EBP$)

The margin of error for a confidence interval of a population proportion, given by the formula $EBP = z_{\alpha/2} \sqrt{\frac{p'q'}{n}}$.

z-score for Proportion

The standardized score for a sample proportion $p'$ calculated using the formula $z = \frac{p' - p}{\sqrt{\frac{pq}{n}}}$.

"Plus-Four" Method

A procedure that adds 4 to the sample size ($n$) and 2 to the count of successes ($x$) before calculating confidence intervals to correct built-in error and produce more accurate results.

Sample Size Formula for Proportion

The formula used to find the required sample size for an error bound $EBP$, calculated as $n = \frac{(z_{\alpha/2})^2 p'q'}{EBP^2} = \left(\frac{z_{\alpha/2}}{EBP}\right)^2 p'q'$ and always rounded up to the next integer.

Conservative Proportion Estimate ($p' = 0.5$)

The value assigned to $p'$ when no prior estimate is available, which maximizes the product $p'q' = (0.5)(0.5) = 0.25$ to ensure the largest possible required sample size $n$.

Hypothesis

A claim subject to testing using statistics.

Null Hypothesis ($H_0$)

A statement that the value of a population parameter (such as mean or proportion) is equal to some claimed value, which always contains an equal sign ($=, \le, \text{ or } \ge$).

Alternative Hypothesis ($H_a$)

A statement that the population parameter has a value that differs from the null hypothesis, always using one of the strict inequality signs $<, >, \text{ or } \ne$; also denoted by $H_1$.

Type I Error

The decision error that occurs when rejecting the null hypothesis ($H_0$) when $H_0$ is actually true.

Type II Error

The decision error that occurs when failing to reject the null hypothesis ($H_0$) when $H_0$ is actually false.

Significance Level ($\alpha$)

The probability of committing a Type I error, defined formally as $\alpha = P(\text{rejecting } H_0 \text{ when } H_0 \text{ is true})$.

Power of the Test ($1 - \beta$)

The probability of making a correct decision by rejecting a false null hypothesis, defined as $1 - \beta = P(\text{rejecting } H_0 \text{ when } H_0 \text{ is false})$.

"RouTiNe FoR FuN" (RTN FRFN)

A mnemonic device used to remember hypothesis testing errors, where RTN represents "Reject True Null" (Type I error) and FRFN represents "Fail to Reject False Null" (Type II error).

z-test for a Single Population Mean

A hypothesis test for a single population mean $\mu$ using the normal distribution $X \sim N(\mu_X, \sigma_X)$, applied when the population standard deviation $\sigma$ is known and the sample is from a normal population or is sufficiently large.

t-test for a Single Population Mean

A hypothesis test for a single population mean $\mu$ using Student's t-distribution $X \sim t_{df}$, applied when the population standard deviation $\sigma$ is unknown and approximated by the sample standard deviation $s$.

P-value Method

A statistical procedure in hypothesis testing where a calculated probability value is used to decide whether to reject or fail to reject the null hypothesis.

Critical Value Method

A statistical procedure in hypothesis testing where a critical cutoff value is identified and compared against the test statistic to decide whether to reject or fail to reject the null hypothesis.