Basic Statistics and Data Science: Exam Study Guide and Practice Questions

Exam Overview and Structure

This study guide covers the foundational topics in basic statistics and data science, as outlined for MATH 012. It includes exam structure, key concepts, formulas, and practice questions to help students prepare for assessments covering Chapters 1–4.

Exam Format

• Duration: 90 minutes

• Coverage: Chapters 1–4

• Sections: Multiple choice (36 points), Short answer (64 points)

• Calculator: Allowed for basic arithmetic (show detailed work for credit)

• Formula Sheet: Provided during the exam

Chapter 1: Introduction to Statistics

Key Concepts

• Statistics: The science of collecting, analyzing, interpreting, and presenting data.

• Population: The entire group of individuals or items under study.

• Sample: A subset of the population selected for analysis.

• Variable: A characteristic or attribute that can assume different values.

• Descriptive Statistics: Methods for summarizing and organizing data.

• Inferential Statistics: Methods for making predictions or inferences about a population based on sample data.

Sampling Methods

• Simple Random Sampling

• Stratified Sampling

• Cluster Sampling

• Systematic Sampling

Additional info: Sampling methods are crucial for ensuring representative data and minimizing bias.

Chapter 2: Types of Data and Data Visualization

Types of Variables

• Categorical (Qualitative): Data that can be grouped into categories (e.g., gender, color).

• Numerical (Quantitative): Data that can be measured or counted (e.g., height, age).

• Discrete: Countable values (e.g., number of students).

• Continuous: Any value within a range (e.g., weight, temperature).

Data Tables and Frequency Distributions

• Construct and interpret frequency tables for categorical and numerical data.

• Use bar charts, Pareto charts, and pie charts for categorical data.

• Use histograms, dot plots, and stem-and-leaf plots for numerical data.

Graphs of Numerical Data

• Histogram: Displays the distribution of numerical data using bars.

• Dot Plot: Shows individual data points.

• Boxplot: Summarizes data using quartiles and identifies outliers.

Measures of Central Tendency

• Mean (Average): $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$

• Median: The middle value when data are ordered.

• Mode: The most frequently occurring value.

Measures of Variability

• Variance: $s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$

• Standard Deviation: $s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}$

• Range: Difference between maximum and minimum values.

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

Empirical Rule (68-95-99.7 Rule)

• For bell-shaped distributions:

• ~68% of data within 1 standard deviation of the mean

• ~95% within 2 standard deviations

• ~99.7% within 3 standard deviations

z-score

• Measures how many standard deviations a value is from the mean.

• Formula: $z = \frac{\text{measurement} - \bar{x}}{s}$

• Used to detect outliers (typically, $|z| > 2$ or $|z| > 3$ indicates a potential outlier).

Linear Transformations

• Transforming data using $y = a + bx$ affects mean and standard deviation.

• Mean: $\bar{y} = a + b\bar{x}$

• Standard deviation: $s_y = |b|s_x$

Chapters 3 & 4: Relationships Between Variables

Contingency Tables

• Used to summarize the relationship between two categorical variables.

• Conditional proportions help identify associations.

Scatterplots

• Graphical representation of the relationship between two quantitative variables.

• Can reveal patterns, trends, and potential outliers.

Correlation Coefficient ($r$)

• Measures the strength and direction of a linear relationship between two variables.

• Formula: $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y}$

• Range: $-1 \leq r \leq 1$

• Correlation does not imply causation.

Linear Regression

• Models the relationship between a response variable $y$ and an explanatory variable $x$.

• Regression line: $\hat{y} = b_0 + b_1x$

• Slope: $b_1 = \frac{S_{xy}}{S_{xx}}$

• Intercept: $b_0 = \bar{y} - b_1\bar{x}$

• Coefficient of determination ($R^2$): Proportion of variance in $y$ explained by $x$.

• Residual: Difference between observed and predicted value ($e = y - \hat{y}$)

Simpson's Paradox

• When a trend appears in several groups of data but disappears or reverses when the groups are combined.

Experimental vs. Observational Studies

• Experimental Study: Researcher manipulates variables and controls conditions.

• Observational Study: Researcher observes without intervention.

Bias and Randomization

• Types of bias: sampling bias, confounding bias, non-response bias, response bias.

• Randomization helps ensure representative samples and valid inferences.

• Double-blind: Neither participants nor experimenters know group assignments.

Practice Questions and Worked Examples

Sample Multiple Choice Questions

1. Definition and identification of statistics, variables, and samples.

2. Interpretation of bar graphs, histograms, and Pareto charts.

3. Understanding sample variance and its properties.

4. Boxplot interpretation and what can/cannot be displayed.

5. Classification of variables as numerical/continuous or categorical/discrete.

6. Properties of bivariate data and correlation.

7. Interpretation of histograms and measures of central tendency.

8. Types of bias in survey questions.

9. Properties of the correlation coefficient $r$.

10. Application of linear transformations and calculation of mean and standard deviation.

Worked Example: Contingency Table

The following table summarizes computer sales by age group and computer type:

Age group	PC	Mac	Total
20-30	70	30	100
30-40	50	50	100
Total	120	80	200

Additional info: Conditional proportions and associations can be calculated from this table to determine if age group and computer type are associated.

Worked Example: Five-Number Summary and Boxplot

• Given data: min = 6, max = 27, Q1, Q2, Q3 to be calculated.

• Five-number summary: Minimum, Q1, Median (Q2), Q3, Maximum

• IQR: $IQR = Q_3 - Q_1$

• Outliers: Any value below $Q_1 - 1.5 \times IQR$ or above $Q_3 + 1.5 \times IQR$

Worked Example: z-score and Outlier Detection

• z-score formula: $z = \frac{x - \bar{x}}{s}$

• Interpretation: z-scores beyond ±2 or ±3 may indicate outliers.

Worked Example: Linear Regression and Correlation

• Regression equation: $y = b_0 + b_1x$

• Interpret slope: Change in $y$ for a one-unit increase in $x$.

• Calculate predicted value and residual for a given $x$.

• Correlation coefficient $r$ calculation using sample data.

Key Formulas Reference

• Sample Mean: $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$

• Sample Variance: $s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$

• z-score: $z = \frac{x - \bar{x}}{s}$

• Correlation Coefficient: $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y}$

• Regression Line: $\hat{y} = b_0 + b_1x$

• Slope: $b_1 = \frac{S_{xy}}{S_{xx}}$

• Intercept: $b_0 = \bar{y} - b_1\bar{x}$

Additional info: Mastery of these concepts and formulas is essential for success in introductory statistics and data science courses.