Course Code:                                       MATH 2275

Course Title:                                        Statistics   I

Course Type:                                       Core

Level:                                                  2

Semester:                                            2

No. of Credits:                                      3

Pre-requisite(s):                                   MATH 2274

 

Course Rationale

Statistics finds applications in a very wide range of fields and is now an important element of general education. Students majoring in mathematics ought, therefore, to have more than a nodding acquaintance with statistics. This introductory course introduces the basic ideas of data analysis, statistical inference and basic statistical methods.

Students will be expected to use statistical software from day one, but a sound knowledge of statistical theory will also be expected. The content of the pre-requisite in probability theory will definitely be used.

It should be noted that the Ministry of Education insists that teachers include such a course in their degree, otherwise they are not paid as graduate teachers.

 

Course Description

The course is a survey of the major ideas of inference, experimental design and statistical methods. The course may be viewed as consisting of three closely connected parts. In the first section, students are introduced to the basics of the statistical packages Minitab and R and their use in descriptive statistics. Emphasis is placed on the use of real data and both summary statistical measures and graphical descriptive devices for continuous and discrete data are discussed.

In the second section, we discuss the frequentist theory of inference, including point estimation, confidence intervals and hypothesis testing. Section three is devoted to various statistical methods. The major ones are regression models and the use of ANOVA in designed experiments. Several of the important basic designs are discussed. We also discuss methods for the analysis of discrete data, such as in contingency tables, and non-parametric procedures.

A knowledge of Probability Theory I is assumed. This is needed since we derive the distributions of most statistics that are used and also discuss systematic mathematical methods for finding point estimators and constructing tests.

Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through weekly assignments and problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures. Assessment will be based on the weekly assignments and in-course tests followed by a final examination based on the whole course

 

Learning Outcomes

Upon successful completion of the course, students will be able to:

  • Use Statistical software, including  R, and Minitab to make summaries of data.
  • Define the terms point estimator, parameter space, bias, mean squared error.
  • Given the population distribution, calculate the MSE of an estimator of a parameter.
  • Derive the maximum likelihood estimator of one or more parameters.
  • Derive confidence intervals of means of normal populations and of proportions.
  • Calculate confidence intervals of means, differences of means, variances of normalpopulations and confidence intervals of proportions.
  • Define the terms statistical hypothesis, null hypothesis, alternative hypothesis, Type 1 error, Type II error, significance level and power of a test.
  • Calculate the significance level and power of a test from a knowledge of the critical region, the null and alternative hypotheses and the population distribution.
  • Perform tests (Z and t) on the means of normal populations and tests on proportions.
  • Test goodness of fit using the chi-squared test.
  • Test independence and equality of proportions in a contingency table
  • Derive the least squares estimates of parameters in a linear regression model.
  • Construct confidence intervals and perform tests on the coefficients of a regression model.
  • Allocate experimental units to treatments in a completely randomized design, to treatments and blocks in a randomized complete block design and Latin Square and to cells in a factorial design.
  • Perform ANOVA in experiments involving one and two factors.

Content

  • Introduction to R and Minitab: A brief introduction to the software packages and to their use in describing and summarizing data involving one variable and several variables using basic statistics, graphs and plots; nominal, ordinal and `interval’ or continuous data will be considered.
  • Sampling Distributions: Distribution of the sample mean and sample variance including the special case of normality; the chi-squared, t and F distributions.
  • Point Estimation: Definitions of parameter, parameter space, point estimator, bias and mean squared error; the theorem MSE= Variance(estimator) + Bias squared; maximum likelihood estimators of one or more parameters.
  • Interval Estimators: The t and F distributions; derivation and calculation of confidence intervals of the means, difference between two means and variances in samples from normal populations with variance known and with variance unknown; confidence intervals for binomial proportions; sample size determination.
  • Hypothesis Testing: Definitions of statistical hypothesis, null and alternative hypothesis, Type I and Type II errors, significance level and power of a test; calculating significance level and power of a test given the critical or rejection region; testing hypotheses concerning the means and variances of normal populations; testing hypotheses concerning proportions; definition and calculation of p values.
  • Contingency Tables: Testing for goodness of fit; independence.
  • Experimental Design: Designed experiments and observational studies; the Completely Randomized Design; One-Way ANOVA; Duncan’s Multiple Range Test; examining assumptions of the linear model; the Randomized Complete Block Design; the statistical model and Two-way ANOVA; Latin Squares; Factorial Designs involving two factors.
  • Regression Analysis: The idea of regression; the method of least squares; simple linear regression; use of graphical techniques to examine assumptions of the linear model; basic estimation, testing and forecasting problems in regression.
  • Non-Parametric methods based on ranks: The sign Test, signed rank test, rank-sum test, Kruskal-Wallis test.

Teaching Methodology

Lectures, tutorials, assignments, statistics labs and problem papers.

Lectures: Three (3) lectures each week (50 minutes each).

Tutorials: One (1) weekly tutorial session or one two-hour lab per week.

Assignments: One assignment (marked) per week.

Additional problems will be given during lectures and tutorials but will not be marked. However, students will need to do some of these, as well as the assignments, in order to learn the material properly and to adequately prepare for examinations and quizzes.

 

Assessment

Coursework: 50% (based on in-term examinations, quizzes, assignments including problems using Minitab) consisting of: Tests: 40%; assignments and data analysis: 10%

Final Examination: 50% (one 2-hour written paper)

 

Course Calendar

 

WEEK

TOPIC

ASSIGNMENT

ASSESSMENT

1

Introduction/Course Overview

Introduction to R; basic descriptive Statistics; definition and distribution of the sample mean and sample variance; the chi-squared, t and F distributions; point estimation; definition of parameter and parameter space; bias and mean squared error

 

Assignment 1 handed out

 

None

2

Maximum likelihood estimation; confidence intervals for mean of a normal population (variance known and unknown)

Assignment 1 returned.

Assignment 2 handed out

Quiz 1

(On Week 1)

3

Sample size; confidence intervals for difference between two means; confidence intervals for the variance and ratio of variances in normal populations; confidence intervals for proportions

 

Quiz 2

(on Week 2)

4

Statistical hypotheses; null and alternative hypotheses; Type I and Type II errors; significance level and power; the Neyman-Pearson approach to hypothesis testing; testing hypotheses involving mean of one normal population

 Assignment 2 returned; Assignment 3

None

5

Testing for differences between means of two normal populations; tests involving the variance of a normal population; testing proportions

 Assignment 4

Quiz 3

Significance level and power of a test

6

Testing goodness of fit and independence;

Assignment 4 returned

None

7

the completely randomized design.

The randomized complete block design.

Assignment 5

Assignment  4 returned

Quiz 4

8

Latin squares; Factorial designs

Assignment 6

Assignment 5 returned

Test 1: Weeks 1-6

9

 

Regression analysis; the method of least squares; the simple linear regression model;

Assignment 7

Quiz 5: Experimental Designs

10

 Graphical methods for checking model assumptions;

Assignment 8

Assignment 7 returned

Quiz 6

11

The Neyman –Pearson lemma; non-parametric methods

Assignment 9;

Assignment 8 returned

Quiz 8: Regression Analysis

12

Non-parametric methods

 Correlation; the sample correlation coefficient

Assignment 10

Assignment 9 returned

Test 2: Experimental Design and Regression

13

Revision

Assignment 10 returned

Revision

None

 

Required Reading

Essential Text:

Probability and Statistical Inference (8th Edition) Prentice-Hall, R. V. Hogg & E. Tanis, 2009.

Other Recommended Texts:

Probability and Statistics for Engineering and the Sciences, J. L. Devore, 2011.