MATH 1160 – Introduction to Statistics: Course Structure and Key Topics

Course Overview

Introduction

This course provides a comprehensive introduction to statistics, covering both theoretical concepts and practical applications. Students will learn about data collection, descriptive and inferential statistics, probability, random variables, sampling distributions, estimation, hypothesis testing, correlation, regression, and chi-square analysis.

• Course Title: MATH 1160 – Introduction to Statistics

• Instructor: Alan Meichsner

• Textbook: Stats: Data and Models, Fourth Canadian Edition, De Veaux et al, Pearson, 2022

• Calculator: Statistical functionality recommended (TI-83+ or TI-84+ available in Math Lab)

Course Structure and Assessment

Grading Breakdown

Grades are determined by a combination of tutorials, homework, quizzes, a midterm, and a final exam. The following table summarizes the grade components:

Component	Frequency	Weight
Tutorials	Weekly	10%
Homework	Weekly	10%
Quizzes	4 per term	20%
Midterm Exam	March 5th	25%
Final Exam	Date TBD	35%

Letter Grade Scale

Final grades are rounded to the nearest percent and assigned according to the following scale:

Percentage	Letter Grade
90 – 100	A+
85 – 89	A
80 – 84	A-
77 – 79	B+
73 – 76	B
70 – 72	B-
65 – 69	C+
60 – 64	C
55 – 59	C-
50 – 54	D
0 – 49	F

Key Course Topics

Descriptive Statistics

Descriptive statistics involve methods for summarizing and displaying data. This includes measures of central tendency (mean, median, mode) and measures of variation (range, variance, standard deviation).

• Central Tendency: Mean, median, mode

• Variation: Range, variance, standard deviation

• Data Display: Histograms, bar charts, boxplots

• Example: Calculating the mean and standard deviation for a set of exam scores

Categorical Data

Categorical data refers to variables that can be divided into groups or categories. Analysis includes frequency tables and graphical displays such as bar charts and pie charts.

• Frequency Tables: Summarize counts for each category

• Bar Charts: Visual representation of categorical data

• Example: Survey responses categorized by gender or preference

Sampling Techniques and Experiments

Sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the whole population. Experiments involve manipulating variables to observe effects.

• Random Sampling: Each member has an equal chance of selection

• Stratified Sampling: Population divided into subgroups, samples taken from each

• Experiments: Control and treatment groups, random assignment

• Example: Randomly selecting students to participate in a study

Probability

Probability quantifies the likelihood of events. Fundamental concepts include sample spaces, events, and probability rules.

• Sample Space: All possible outcomes

• Probability Rules: Addition and multiplication rules

• Example: Calculating the probability of drawing an ace from a deck of cards

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

Random Variables and Distributions

Random variables assign numerical values to outcomes of random phenomena. Distributions describe the probabilities associated with each value.

• Discrete Random Variables: Take on specific values (e.g., binomial)

• Continuous Random Variables: Take on any value within an interval (e.g., normal)

• Example: Number of heads in 10 coin tosses (binomial random variable)

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

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve. It is defined by its mean and standard deviation.

• Properties: Symmetrical, mean = median = mode

• Standard Normal: Mean 0, standard deviation 1

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

• Example: Heights of adult males

Sampling Distributions and Central Limit Theorem

Sampling distributions describe the distribution of a statistic (e.g., sample mean) over repeated samples. The Central Limit Theorem states that the sampling distribution of the sample mean approaches normality as sample size increases.

• Central Limit Theorem: For large n, sample mean is approximately normal

• Formula: $\bar{x} \sim N(\mu, \frac{\sigma}{\sqrt{n}})$

• Example: Average test scores from repeated samples

Estimation and Confidence Intervals

Estimation involves using sample data to estimate population parameters. Confidence intervals provide a range of plausible values for the parameter.

• Point Estimate: Single value estimate (e.g., sample mean)

• Confidence Interval: Range with specified probability

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

• Example: Estimating average income with 95% confidence

Hypothesis Testing

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

• Null Hypothesis (H0): Statement of no effect

• Alternative Hypothesis (HA): Statement of effect or difference

• Test Statistic: Measures evidence against H0

• Formula: $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

• Example: Testing if a new drug is effective

Inference for Proportions and Means

Statistical inference allows for conclusions about population proportions and means based on sample data.

• One Proportion: Confidence intervals and hypothesis tests for a single proportion

• Two Proportions: Comparing proportions between groups

• One Mean: Inference for population mean

• Two Means: Comparing means between groups

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

• Example: Comparing average scores between two classes

Correlation and Regression

Correlation measures the strength and direction of linear relationships between variables. Regression models the relationship between a dependent variable and one or more independent variables.

• Scatterplots: Visualize relationships

• Correlation Coefficient: $r$ measures linear association

• Linear Regression: Predicts values using a linear equation

• Formula: $y = a + bx$

• Example: Predicting sales based on advertising budget

Chi-Square Analysis

Chi-square tests are used for categorical data to assess goodness of fit and independence.

• Goodness of Fit: Tests if observed frequencies match expected

• Test for Independence: Assesses association between categorical variables

• Formula: $\chi^2 = \sum \frac{(O - E)^2}{E}$

• Example: Testing if gender and preference are independent

Course Policies and Resources

Attendance and Academic Integrity

• Attendance: Required; missing more than 30% results in a non-attendance grade (UN)

• Academic Integrity: Violations handled per college policy

• Calculator Policy: Memories cleared before tests

Math Lab and Tutorials

• Math Lab: Room S3910; provides homework help and concept review

• Tutorials: Weekly sessions for group problem solving

Tentative Lecture Schedule

Weekly Topics

Week	Date	Topic	Chapters/Sections
1	Jan 6	Introduction – data, sampling techniques, experiments	1, 9, 10
	Jan 8	Categorical data	2
2	Jan 13	Numerical data part 1	3
	Jan 15	Numerical data part 2	3
3	Jan 20	Probability part 1	11
	Jan 22	Probability part 2 (Quiz 1)	12
4	Jan 27	Probability part 3	12
	Jan 29	Random variables	13.1 – 13.3
5	Feb 3	Normal distributions	5.1 – 5.4
	Feb 5	Normal random variables (Quiz 2)	13.6
6	Feb 10	Binomial random variables	13.4, 13.7, 13.8
	Feb 12	Sampling distributions, Central Limit Theorem	14
7	Feb 24	Confidence intervals for one proportion	15
	Feb 26	More on confidence intervals / Intro to hypothesis tests	15, 16
8	Mar 3	More on hypothesis tests for one proportion	16, 17
	Mar 5	Midterm Exam
9	Mar 10	Inference for two proportions	21
	Mar 12	Inference for one mean	18
10	Mar 17	Inference for two means	19
	Mar 19	Inference for matched paired	20
11	Mar 24	Scatter plots and correlation (Quiz 3)	6
	Mar 26	Linear regression and residuals	7, 8
12	Mar 31	More on Linear regression and residuals	7, 8
	Apr 2	Goodness of fit	22.1
13	Apr 7	Chi square test for independence (Quiz 4)	22.4
	Apr 9	Catch-up / Review
14	Apr 14	Catch-up / Review
	Final Exam Period	April 16th – 24th

Summary

This syllabus outlines a foundational statistics course, covering all major topics required for college-level statistics. Students are expected to engage with weekly assignments, tutorials, quizzes, and exams, and to utilize resources such as the Math Lab for additional support. The course emphasizes both theoretical understanding and practical application of statistical methods.