STA1502
EXAM PACK
2023
QUESTIONS
AND
ANSWERS
, i STA1502/1
CONTENTS
Orientation iii
STUDY UNIT 1
1.1 Introduction 1
1.2 Inference about the Difference Between Two Population Means: 1
Independent Samples
1.3 Observational and Experimental Data 18
1.4 Inference about the Difference Between Two Population Means: 19
Matched Pairs Experiment
1.5 Inference about the Ratio of Two Variances 28
1.6 Self-correcting Exercises for Unit 1 38
1.7 Solutions to Self-correcting Exercises for Unit 1 39
1.8 Learning Outcomes 46
STUDY UNIT 2
2.1 Introduction 48
2.2 Inference about the Difference Between Two Population Proportions 48
2.3 One-Way Analysis of Variance 57
2.4 Multiple Comparisons 68
2.5 Analysis of Variance experimental designs (read only) 80
2.6 Randomized Block(two-way) Analysis of Variance 81
2.7 Self-correcting Exercises for Unit 2 87
2.8 Solutions to Self-correcting Exercises for Unit 2 89
2.9 Learning Outcomes 98
STUDY UNIT 3
3.1 Chi–square test 100
3.2 Chi-squared goodness-of-fit test 101
3.3 Chi-squared test of a Contingency Table 107
3.4 Summary of test on nominal data 109
STUDY UNIT 4
4.1 Simple linear regression and correlation 118
4.2 Simple Linear Regression and Correlations 118
4.3 Diagnostic Tools for checking the regressionassumptions 122
STUDY UNIT 5
5.1 Non parametric statistics 131
5.2 The Wilcoxon Rank Sum Test: Independent Random Samples 131
, ii
5.3 Sign Test and Wilcoxon Signed Rank Sum Test 137
5.4 The Wilcoxon Signed Rank Sum Test for a matched paired experiment 147
5.5 The Kruskal–Wills H –test for completely randomized designs 151
5.6 Friedman Test for the Randomized Block Design 154
STUDY UNIT 6
6.1 Introduction 158
6.2 Components of time series 158
6.3 Smoothing techniques 160
6.4 Trend and seasonal effects 164
6.5 Introduction to forecasting 167
6.6 Forcasting models 167
, iii STA1502/1
ORIENTATION
Welcome
Welcome to STA1502. This module is the second one of the first-year statistics courses. STA1501
and STA1502 form the first year Statistics course for students from the College of Economic and
Management Sciences. If you are a BSc student in the College of Science, Engineering and
Technology, the three modules STA1501 and STA1502 and STA1503 form the first year in Statistics.
In the preceding module STA1501, we treated probability and probability distributions, and unless
one has a proper understanding of the laws of probability, the mechanisms underlying statistical data
analysis will not be understood properly. Probability theory is the tool that makes statistical inference
possible. In STA1502, we consider to the applications of the probability distributions. You have
learned in STA1501 that the shape of the normal distribution is determined by the value of the mean
2;
and the variance whilst the shape of the binomial distribution is determined by the sample size
n and the probability of a success p. These critical values are called parameters. We most often
don’t know what the values of the parameters are and thus we cannot "utilise" these distributions (i.e.
use the mathematical formula to draw a probability density graph or compute specific probabilities)
unless we somehow estimate these unknown parameters. It makes perfect logical sense that to
estimate the value of an unknown population parameter, we compute a corresponding or comparable
characteristic of the sample.
The objective of this module is to focus on the issues related to prediction and inference in statistics
and therefore it is called Statistical Inference and the "I" in the title indicates that it is a module at
the first level. We draw inference about a population (a complete set of data) based on the limited
information contained in a sample. In dictionary terms, inference is the act or process of inferring;
to infer means to conclude or judge from premises or evidence; meaning to derive by reasoning.
In general, the term implies a conclusion based on experience or knowledge. More specifically in
statistics, we have as evidence the limited information contained in the outcome of a sample and
we want to conclude something about the unknown population from which the sample was drawn.
The set of principles, procedures and methods that we use to study populations by making use of
information obtained from samples is called statistical inference.
Learning outcomes
There are very specific outcomes for this module, listed below. Throughout your study of this module
you must come back to this page, sit back and reflect upon them, think them through, digest them
into your system and feel confident in the end that you have mastered the following outcomes:
EXAM PACK
2023
QUESTIONS
AND
ANSWERS
, i STA1502/1
CONTENTS
Orientation iii
STUDY UNIT 1
1.1 Introduction 1
1.2 Inference about the Difference Between Two Population Means: 1
Independent Samples
1.3 Observational and Experimental Data 18
1.4 Inference about the Difference Between Two Population Means: 19
Matched Pairs Experiment
1.5 Inference about the Ratio of Two Variances 28
1.6 Self-correcting Exercises for Unit 1 38
1.7 Solutions to Self-correcting Exercises for Unit 1 39
1.8 Learning Outcomes 46
STUDY UNIT 2
2.1 Introduction 48
2.2 Inference about the Difference Between Two Population Proportions 48
2.3 One-Way Analysis of Variance 57
2.4 Multiple Comparisons 68
2.5 Analysis of Variance experimental designs (read only) 80
2.6 Randomized Block(two-way) Analysis of Variance 81
2.7 Self-correcting Exercises for Unit 2 87
2.8 Solutions to Self-correcting Exercises for Unit 2 89
2.9 Learning Outcomes 98
STUDY UNIT 3
3.1 Chi–square test 100
3.2 Chi-squared goodness-of-fit test 101
3.3 Chi-squared test of a Contingency Table 107
3.4 Summary of test on nominal data 109
STUDY UNIT 4
4.1 Simple linear regression and correlation 118
4.2 Simple Linear Regression and Correlations 118
4.3 Diagnostic Tools for checking the regressionassumptions 122
STUDY UNIT 5
5.1 Non parametric statistics 131
5.2 The Wilcoxon Rank Sum Test: Independent Random Samples 131
, ii
5.3 Sign Test and Wilcoxon Signed Rank Sum Test 137
5.4 The Wilcoxon Signed Rank Sum Test for a matched paired experiment 147
5.5 The Kruskal–Wills H –test for completely randomized designs 151
5.6 Friedman Test for the Randomized Block Design 154
STUDY UNIT 6
6.1 Introduction 158
6.2 Components of time series 158
6.3 Smoothing techniques 160
6.4 Trend and seasonal effects 164
6.5 Introduction to forecasting 167
6.6 Forcasting models 167
, iii STA1502/1
ORIENTATION
Welcome
Welcome to STA1502. This module is the second one of the first-year statistics courses. STA1501
and STA1502 form the first year Statistics course for students from the College of Economic and
Management Sciences. If you are a BSc student in the College of Science, Engineering and
Technology, the three modules STA1501 and STA1502 and STA1503 form the first year in Statistics.
In the preceding module STA1501, we treated probability and probability distributions, and unless
one has a proper understanding of the laws of probability, the mechanisms underlying statistical data
analysis will not be understood properly. Probability theory is the tool that makes statistical inference
possible. In STA1502, we consider to the applications of the probability distributions. You have
learned in STA1501 that the shape of the normal distribution is determined by the value of the mean
2;
and the variance whilst the shape of the binomial distribution is determined by the sample size
n and the probability of a success p. These critical values are called parameters. We most often
don’t know what the values of the parameters are and thus we cannot "utilise" these distributions (i.e.
use the mathematical formula to draw a probability density graph or compute specific probabilities)
unless we somehow estimate these unknown parameters. It makes perfect logical sense that to
estimate the value of an unknown population parameter, we compute a corresponding or comparable
characteristic of the sample.
The objective of this module is to focus on the issues related to prediction and inference in statistics
and therefore it is called Statistical Inference and the "I" in the title indicates that it is a module at
the first level. We draw inference about a population (a complete set of data) based on the limited
information contained in a sample. In dictionary terms, inference is the act or process of inferring;
to infer means to conclude or judge from premises or evidence; meaning to derive by reasoning.
In general, the term implies a conclusion based on experience or knowledge. More specifically in
statistics, we have as evidence the limited information contained in the outcome of a sample and
we want to conclude something about the unknown population from which the sample was drawn.
The set of principles, procedures and methods that we use to study populations by making use of
information obtained from samples is called statistical inference.
Learning outcomes
There are very specific outcomes for this module, listed below. Throughout your study of this module
you must come back to this page, sit back and reflect upon them, think them through, digest them
into your system and feel confident in the end that you have mastered the following outcomes:
