Math 105: Basic Probability and Statistics – Course Syllabus and Study Guide

Math 105: Basic Probability and Statistics

Course Overview

This course provides an introduction to fundamental concepts in probability and statistics, focusing on methods useful for decision making and estimation across various fields. The curriculum emphasizes statistical reasoning, conceptual understanding, and practical applications, rather than algorithmic procedures.

• Instructor: Myra Young

• Textbook: Basic Practice of Statistics, 9th edition, by Moore and Notz

• Chapters Covered: 1–6, 8–9, 15–18, 20–22, 26

Learning Objectives

• Understand statistical methods for decision making and estimation.

• Develop statistical and probabilistic reasoning skills.

• Recognize unifying themes and concepts in statistics.

• Build a foundation for future statistical applications.

• Focus on conceptual understanding over algorithmic processes.

Topics Covered

Graphical Presentation of Data

Students will learn to construct and interpret elementary graphical displays, which are essential for visualizing and summarizing data.

• Stemplots: Display quantitative data to show distribution.

• Histograms: Show frequency distribution of numerical data.

• Boxplots: Summarize data using quartiles and highlight outliers.

• Interpretation: Identify patterns, trends, and anomalies in data.

• Example: Creating a histogram for exam scores to visualize performance distribution.

Numerical Measures of Data

Descriptive statistics provide measures of central tendency, spread, and relative standing, which are fundamental for summarizing data sets.

• Measures of Center: Mean, median, mode.

• Measures of Spread: Range, variance, standard deviation.

• Relative Standing: Percentiles, quartiles, z-scores.

• Example: Calculating the mean and standard deviation for a set of survey responses.

Normal Distributions

The normal distribution is a key concept in statistics, used to model many natural phenomena and underpinning many inferential methods.

• Definition: A symmetric, bell-shaped distribution characterized by its mean and standard deviation.

• Applications: Standardizing scores, probability calculations.

• Limitations: Not all data sets are normally distributed.

• Formula:

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$

• Example: Using z-scores to determine the probability of a value occurring within a normal distribution.

Sampling Designs and Experimental Designs

Understanding how data is collected is crucial for interpreting results and avoiding bias.

• Random Errors: Variability due to chance.

• Nonrandom Errors: Systematic bias from flawed sampling or measurement.

• Sampling Designs: Simple random sampling, stratified sampling, cluster sampling.

• Experimental Designs: Use of blocking to control for confounding variables.

• Example: Designing a survey to minimize bias and maximize representativeness.

Sampling Distributions and the Central Limit Theorem

Sampling distributions describe the distribution of a statistic (e.g., mean) across repeated samples, and the Central Limit Theorem explains why sample means tend to be normally distributed.

• Sampling Distribution: Distribution of a statistic over many samples.

• Central Limit Theorem: For large samples, the sampling distribution of the mean is approximately normal, regardless of the population distribution.

• Formula:

$\text{If } n \text{ is large, } \bar{x} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$

• Example: Estimating the average height of students using repeated samples.

Confidence Intervals

Confidence intervals provide a range of plausible values for population parameters, based on sample data.

• For Population Mean: Based on large or small samples.

• For Population Proportion: Based on large samples.

• Formula for Mean (large sample):

$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

• Formula for Proportion:

$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Example: Calculating a 95% confidence interval for the average test score.

Significance Testing

Statistical tests assess whether observed data provide evidence against a null hypothesis.

• Single Mean or Proportion: Test hypotheses about population parameters.

• Types of Errors: Type I (false positive), Type II (false negative).

• Level of Significance: Chosen based on error type concern (commonly 0.05).

• Example: Testing whether a new teaching method improves average scores.

Inference for Two Samples

Comparing two populations using sample data to infer differences in means or proportions.

• Comparison of Means: Use t-tests for independent samples.

• Formula:

$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Example: Comparing average scores between two classes.

Descriptive Analysis of Categorical Data

Categorical data analysis includes summarizing and interpreting data in categories, and understanding phenomena like Simpson’s paradox.

• Simpson’s Paradox: A trend appears in several groups but reverses when groups are combined.

• Example: Analyzing pass rates by gender and overall.

Correlation and Simple Linear Regression (SLR)

Correlation measures the strength and direction of linear relationships, while regression models the relationship between variables.

• Correlation: Quantifies linear association (Pearson’s r).

• Simple Linear Regression: Models the relationship between two variables.

• Regression Equation:

$y = \beta_0 + \beta_1 x + \varepsilon$

• Residual Standard Error: Measures the typical size of residuals.

• Coefficient of Determination (R2): Proportion of variance explained by the model.

• Inference for Slope: Test whether the slope is significantly different from zero.

• Prediction Intervals: Estimate the range for individual predictions or mean response.

• Example: Predicting final exam scores based on homework scores.

Course Structure and Assessment

Grading Breakdown

Category	Percentage
Attendance	10%
Homework	25%
Exam #1	20%
Exam #2	20%
Final Exam	25%
Total	100%

Grading Rubric

Grade	Percentage Range
A	93 - 100
A-	90 – 92.9999
B+	87 – 89.9999
B	83 – 86.9999
B-	80 – 82.9999
C+	77 – 79.9999
C	73 – 76.9999
C-	70 – 72.9999
D+	67 – 69.9999
D	60 – 66.9999
F	Below 60

Policies and Advice

• Homework: Assignments are frequent and time-consuming; incremental work is recommended.

• Exams: Two midterms and a cumulative final; strict policy on missed exams and documentation.

• Attendance: Points deducted for absences or lateness; attendance is critical for success.

• Academic Integrity: Cheating results in severe penalties; collaboration is allowed only with substantive participation.

• Accommodations: Students with disabilities should contact Disability Services for support.

• Diversity and Inclusion: The course supports an inclusive and respectful learning environment.

• Provision for Needs: Students facing financial or personal challenges are encouraged to seek support.

Additional info:

• Topics align closely with standard introductory statistics chapters, including graphical data presentation, descriptive statistics, normal distributions, sampling, confidence intervals, hypothesis testing, comparison of means/proportions, and regression.

• Some advanced topics (e.g., ANOVA, chi-square tests) may be covered if time allows, as inferred from the textbook chapters listed.