STAT C1000E: Course Overview and Key Concepts in Statistics

Course Overview: Introduction to Statistics

Course Description

This course provides a comprehensive introduction to statistics, covering both descriptive and inferential methods. Students will learn about probability, measures of central tendency and dispersion, sampling, estimation, hypothesis testing, analysis of variance, chi-square tests, Student's t-distribution, linear correlation, and regression analysis. The course also includes practical applications using Excel spreadsheets.

• Descriptive Statistics: Summarizing and describing data using tables, graphs, and numerical measures.

• Inferential Statistics: Drawing conclusions about populations based on sample data.

• Probability: Understanding the likelihood of events and distributions.

• Statistical Distributions: Including normal, Student's t, and chi-square distributions.

• Hypothesis Testing: Procedures for testing claims about population parameters.

• Estimation: Determining population parameters and sample sizes.

• Analysis of Variance (ANOVA): Comparing means across multiple groups.

• Correlation and Regression: Analyzing relationships between variables.

Course Learning Objectives

Student Learning Outcomes (SLOs)

• SLO 1: Test hypotheses for sample proportion, mean, and standard deviation.

• SLO 2: Determine the probability of an event in a given distribution.

• SLO 3: Perform correlation and linear regression analysis.

Main Topics Covered

1. Introduction to Statistics

Statistics is the science of collecting, analyzing, interpreting, and presenting data. It is divided into two main branches: descriptive and inferential statistics.

• Descriptive Statistics: Methods for summarizing data.

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

2. Exploring Data with Tables and Graphs

Data can be organized and visualized using frequency tables, histograms, bar charts, and scatterplots.

• Frequency Table: Shows how often each value occurs.

• Histogram: Visual representation of data distribution.

• Scatterplot: Used to examine relationships between two variables.

3. Describing, Exploring, and Comparing Data

Key measures include central tendency (mean, median, mode) and dispersion (range, variance, standard deviation).

• Mean: The average value.

• Median: The middle value when data is ordered.

• Mode: The most frequent value.

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

• Standard Deviation: Square root of variance; formula: $s = \sqrt{s^2}$

4. Probability

Probability quantifies the likelihood of events. It is foundational for understanding statistical inference.

• Probability of an event: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

• Probability distributions: Describe how probabilities are distributed over values.

5. Discrete Probability Distributions

Discrete distributions include the binomial and Poisson distributions, which model countable outcomes.

• Binomial Distribution: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Poisson Distribution: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$

6. Normal Probability Distributions

The normal distribution is a continuous, symmetric distribution important in many statistical analyses.

• Normal Distribution Formula: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

• Standard Normal: $Z = \frac{X - \mu}{\sigma}$

7. Estimating Parameters and Determining Sample Sizes

Estimation involves using sample data to estimate population parameters, such as the mean or proportion.

• Confidence Interval for Mean: $\bar{x} \pm z^* \frac{s}{\sqrt{n}}$

• Sample Size Formula: $n = \left( \frac{z^* s}{E} \right)^2$

8. Hypothesis Testing

Hypothesis testing is a formal procedure for testing claims about population parameters.

• Null Hypothesis ($H_0$): The default assumption.

• Alternative Hypothesis ($H_a$): The claim to be tested.

• Test Statistic: $z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$

• p-value: Probability of observing the data if $H_0$ is true.

9. Inferences from Two Samples

Comparing two samples to determine if there is a significant difference between their means or proportions.

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

10. Correlation and Regression

Correlation measures the strength and direction of a linear relationship between two variables. Regression models the relationship.

• Correlation Coefficient: $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

• Linear Regression Equation: $y = a + bx$

11. Goodness-of-Fit and Contingency Tables

Chi-square tests are used to assess how well observed data fit expected distributions and to analyze categorical data.

• Chi-square Test Statistic: $\chi^2 = \sum \frac{(O - E)^2}{E}$

• Contingency Table: Used to examine relationships between categorical variables.

12. Analysis of Variance (ANOVA)

ANOVA is used to compare means across three or more groups to determine if at least one group mean is different.

• ANOVA F-statistic: $F = \frac{\text{Between-group variance}}{\text{Within-group variance}}$

Assessment and Course Logistics

• Quizzes: Weekly quizzes, lowest score dropped.

• Final Exam: Scheduled for Wednesday, Jun 4, 2026, 2:30–4:30 pm.

• Excel Usage: Excel spreadsheets may be used for data analysis.

Additional info: The course covers all major topics listed in a standard college statistics curriculum, including probability, distributions, hypothesis testing, estimation, ANOVA, chi-square, and regression. Students are expected to gain practical skills in data analysis and interpretation.