STAT101 - Introduction to Statistics & Probability Flashcards

Question 1

Mean (Arithmetic Average)

Accepted Answer

The sum of all observed values divided by the total number of observations. Population mean: $\mu = \frac{\sum X_i}{N}$; Sample mean: $\bar{x} = \frac{\sum x_i}{n}$. It is sensitive to extreme values (outliers).

Question 2

Median

Accepted Answer

The middle value when data is sorted in ascending order. For an even sample size $n$, it is the average of the two middle values. Unlike the mean, the median is resistant to outliers.

Question 3

Mode

Accepted Answer

The value that occurs most frequently in a dataset. A dataset can have one mode (unimodal), multiple modes (bimodal/multimodal), or no mode at all if all values occur with equal frequency.

Question 4

Variance

Accepted Answer

A measure of numerical dispersion representing the average squared deviation from the mean. Population variance: $\sigma^2 = \frac{\sum (X_i - \mu)^2}{N}$. Sample variance: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$ (uses $n-1$ for Bessel's correction).

Question 5

Standard Deviation

Accepted Answer

The positive square root of the variance ($s = \sqrt{s^2}$ or $\sigma = \sqrt{\sigma^2}$). It quantifies spread in the original measurement units of the data.

Question 6

Interquartile Range (IQR)

Accepted Answer

The range containing the central 50% of data, calculated as $IQR = Q_3 - Q_1$. Values beyond $Q_1 - 1.5 	imes IQR$ or $Q_3 + 1.5 	imes IQR$ are commonly identified as potential outliers.

Question 7

Bernoulli Distribution

Accepted Answer

A discrete probability distribution for a single trial with two possible outcomes: success ($X = 1$) with probability $p$, and failure ($X = 0$) with probability $1 - p$. Mean: $\mu = p$; Variance: $\sigma^2 = p(1 - p)$.

Question 8

Binomial Distribution

Accepted Answer

Models the number of successes $k$ in $n$ independent Bernoulli trials with constant success probability $p$. PMF: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$. Mean: $\mu = np$; Variance: $\sigma^2 = np(1 - p)$.

Question 9

Poisson Distribution

Accepted Answer

Models the number of events occurring within a fixed interval of time or space, given a constant rate $\lambda$. PMF: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$. Mean: $\mu = \lambda$; Variance: $\sigma^2 = \lambda$.

Question 10

Normal Distribution

Accepted Answer

A continuous bell-shaped distribution defined by parameters $\mu$ (mean) and $\sigma$ (standard deviation), denoted $N(\mu, \sigma^2)$. Follows the Empirical Rule: 68%, 95%, and 99.7% of data lie within 1, 2, and 3 standard deviations of $\mu$.

Question 11

Standard Normal Distribution & Z-Score

Accepted Answer

A normal distribution with $\mu = 0$ and $\sigma = 1$. Any score $X$ transforms to a $Z$-score via $Z = \frac{X - \mu}{\sigma}$, measuring how many standard deviations $X$ lies from the mean.

Question 12

Central Limit Theorem (CLT)

Accepted Answer

States that the sampling distribution of the sample mean $\bar{x}$ approaches a normal distribution $N\left(\mu, \frac{\sigma^2}{n}\right)$ as sample size $n$ increases (typically $n \ge 30$), regardless of the shape of the underlying population.

Question 13

Null ($H_0$) vs. Alternative ($H_a$) Hypothesis

Accepted Answer

$H_0$ represents the default assumption of no effect, no difference, or status quo. $H_a$ represents the parameter claim being tested for evidence of a significant effect or difference.

Question 14

Significance Level ($\alpha$)

Accepted Answer

The probability threshold chosen prior to testing for rejecting $H_0$. It represents the maximum acceptable risk of committing a Type I error (commonly set at $\alpha = 0.05$).

Question 15

Type I and Type II Errors

Accepted Answer

Type I Error (false positive): Rejecting $H_0$ when $H_0$ is true (Probability $= \alpha$). Type II Error (false negative): Failing to reject $H_0$ when $H_0$ is false (Probability $= \beta$).

Question 16

p-value

Accepted Answer

The probability of observing a test statistic as extreme as, or more extreme than, the sample result assuming $H_0$ is true. If $p	ext{-value} \le \alpha$, reject $H_0$; if $p	ext{-value} > \alpha$, fail to reject $H_0$.

Question 17

Steps in Hypothesis Testing

Accepted Answer

1. State $H_0$ and $H_a$. 2. Set significance level $\alpha$. 3. Compute test statistic (e.g., $z$-score or $t$-score). 4. Calculate the p-value or compare to critical value. 5. Make statistical decision and state conclusion in context.

Question 18

Simple Linear Regression Model

Accepted Answer

Models a linear relationship between an explanatory variable $X$ and response variable $Y$: $Y = \beta_0 + \beta_1 X + \epsilon$, where $\beta_0$ is the intercept, $\beta_1$ is the slope, and $\epsilon \sim N(0, \sigma^2)$ is residual error.

Question 19

Least Squares Method

Accepted Answer

A method to calculate the best-fit line $\hat{Y} = b_0 + b_1 X$ by minimizing the sum of squared residuals $\sum (y_i - \hat{y}_i)^2$. Slope formula: $b_1 = \frac{	ext{Cov}(X, Y)}{	ext{Var}(X)} = r \frac{s_y}{s_x}$.

Question 20

Coefficient of Determination ($R^2$)

Accepted Answer

The proportion of total variance in response variable $Y$ explained by linear regression on predictor $X$. $R^2 = 1 - \frac{SSR}{SST}$, bounded between $0$ and $1$ ($0\% \le R^2 \le 100\%$).

STAT101 - Introduction to Statistics & Probability

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