Mastervincent
HRM3706 SUMMARISED NOTES
STA1610 EXAM
PACK
FOR ASSISTANCE CONTACT
, lOMoARcPSD|51600623
Contents
1 Data and Statistics 1
1.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Elements, variables, and observations . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Scales of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.3 Categorical and Quantitative Data . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 Cross-sectional and time series data . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Statistical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Data acquisition errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Analytics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Big Data and Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Descriptive Statistics: Tabular and Graphical Presentations 9
2.1 Summarising Categorical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Relative Frequency and Percentage Frequency Distributions . . . . . . . . . . . 10
2.1.2 Bar Charts and Pie Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Summarising Quantitative Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Cumulative Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Numerical Measures 13
3.1 Measures of Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Measures of Variability (Dispersion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Measures of Distributional Shape, Relative Location, and Detecting Outliers . . . . . . 16
3.4 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5 Measures of Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Introduction to Probability 19
4.1 Experiments, Counting Rules, and Assigning Probabilities . . . . . . . . . . . . . . . . . 19
4.1.1 Counting Rules, Combinations, and Permutations . . . . . . . . . . . . . . . . . 20
4.1.2 Assigning Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Events and Their Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Some Basic Relationships of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 Objective Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.5 Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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CONTENTS
5 Discrete Probability Distributions 31
5.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Discrete Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3 Expected Value and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.4 Binomial Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.5 Poisson Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Continuous Probability Distributions 45
6.1 Normal Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.1.1 Standard Normal Probability Distribution . . . . . . . . . . . . . . . . . . . . . 46
6.1.2 Computing Probabilities for Any Normal Distribution . . . . . . . . . . . . . . . 46
6.2 Cumulative Standard Normal Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 52
7 Sampling and Sampling Distributions 55
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Sampling Distribution of 𝑋 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3 Sampling Distribution of Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8 Interval Estimation 67
8.1 Population Mean: Standard Deviation (𝜎) Known . . . . . . . . . . . . . . . . . . . . . 67
8.2 Population Mean: Standard Deviation (𝜎) Unknown . . . . . . . . . . . . . . . . . . . . 69
8.3 Confidence Interval Estimate for Proportion . . . . . . . . . . . . . . . . . . . . . . . . 71
9 Hypothesis Testing 73
9.1 Fundamental Concepts of Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 Hypothesis Testing of the Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.2.1 Population Standard Deviation (𝜎) Known . . . . . . . . . . . . . . . . . . . . . 74
9.2.2 Population Standard Deviation (𝜎) Unknown . . . . . . . . . . . . . . . . . . . 75
9.3 Hypothesis Testing for Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
10 Chi-Square Distribution 77
11 Regression and Correlation Analysis 83
11.1 Simple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
11.2 Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
iv
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Chapter 1
Data and Statistics
Statistics
The art and science of collecting, analysing, presenting, and interpreting data.
1.1 Data
Data
Facts and figures collected, analysed, and summarised for presentation and interpretation. All
the data in a particular study are referred to as the data set for the study.
1.1.1 Elements, variables, and observations
Elements are the entities on which data are collected. A variable is a characteristic of interest for the
elements. Measurements collected on each variable for every element in a study provide the data. The
set of measurements obtained for a particular element is called an observation.
1.1.2 Scales of Measurement
A scale of measurement determines the amount of information contained in the data, and indicates
the most appropriate data summarisation and statistical analyses.
Nominal Data for a variable consist of labels or names used to identify an attribute of an element. In
cases where the scale of measurement is nominal, a numeric code as well as non-numeric labels
may be used.
Ordinal The data exhibit the properties of nominal data, and the order or rank of the data is
meaningful. Note that ordinal data can also be recorded using a numeric code.
Interval The data show the properties of ordinal data, and the interval between values is expressed
in terms of a fixed unit of measure. Interval data are always numeric.
Ratio The data show the properties of interval data, and the ratio of two values is meaningful.
Variables such as distance, height, weight, and time use the ratio scale of measurement. This
scale requires that a zero value be included.
1
Downloaded by Vincent kyalo ()
HRM3706 SUMMARISED NOTES
STA1610 EXAM
PACK
FOR ASSISTANCE CONTACT
, lOMoARcPSD|51600623
Contents
1 Data and Statistics 1
1.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Elements, variables, and observations . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Scales of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.3 Categorical and Quantitative Data . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 Cross-sectional and time series data . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Statistical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Data acquisition errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Analytics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Big Data and Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Descriptive Statistics: Tabular and Graphical Presentations 9
2.1 Summarising Categorical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Relative Frequency and Percentage Frequency Distributions . . . . . . . . . . . 10
2.1.2 Bar Charts and Pie Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Summarising Quantitative Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Cumulative Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Numerical Measures 13
3.1 Measures of Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Measures of Variability (Dispersion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Measures of Distributional Shape, Relative Location, and Detecting Outliers . . . . . . 16
3.4 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5 Measures of Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Introduction to Probability 19
4.1 Experiments, Counting Rules, and Assigning Probabilities . . . . . . . . . . . . . . . . . 19
4.1.1 Counting Rules, Combinations, and Permutations . . . . . . . . . . . . . . . . . 20
4.1.2 Assigning Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Events and Their Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Some Basic Relationships of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 Objective Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.5 Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
iii
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CONTENTS
5 Discrete Probability Distributions 31
5.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Discrete Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3 Expected Value and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.4 Binomial Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.5 Poisson Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Continuous Probability Distributions 45
6.1 Normal Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.1.1 Standard Normal Probability Distribution . . . . . . . . . . . . . . . . . . . . . 46
6.1.2 Computing Probabilities for Any Normal Distribution . . . . . . . . . . . . . . . 46
6.2 Cumulative Standard Normal Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 52
7 Sampling and Sampling Distributions 55
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Sampling Distribution of 𝑋 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3 Sampling Distribution of Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8 Interval Estimation 67
8.1 Population Mean: Standard Deviation (𝜎) Known . . . . . . . . . . . . . . . . . . . . . 67
8.2 Population Mean: Standard Deviation (𝜎) Unknown . . . . . . . . . . . . . . . . . . . . 69
8.3 Confidence Interval Estimate for Proportion . . . . . . . . . . . . . . . . . . . . . . . . 71
9 Hypothesis Testing 73
9.1 Fundamental Concepts of Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 Hypothesis Testing of the Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.2.1 Population Standard Deviation (𝜎) Known . . . . . . . . . . . . . . . . . . . . . 74
9.2.2 Population Standard Deviation (𝜎) Unknown . . . . . . . . . . . . . . . . . . . 75
9.3 Hypothesis Testing for Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
10 Chi-Square Distribution 77
11 Regression and Correlation Analysis 83
11.1 Simple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
11.2 Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
iv
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, lOMoARcPSD|51600623
Chapter 1
Data and Statistics
Statistics
The art and science of collecting, analysing, presenting, and interpreting data.
1.1 Data
Data
Facts and figures collected, analysed, and summarised for presentation and interpretation. All
the data in a particular study are referred to as the data set for the study.
1.1.1 Elements, variables, and observations
Elements are the entities on which data are collected. A variable is a characteristic of interest for the
elements. Measurements collected on each variable for every element in a study provide the data. The
set of measurements obtained for a particular element is called an observation.
1.1.2 Scales of Measurement
A scale of measurement determines the amount of information contained in the data, and indicates
the most appropriate data summarisation and statistical analyses.
Nominal Data for a variable consist of labels or names used to identify an attribute of an element. In
cases where the scale of measurement is nominal, a numeric code as well as non-numeric labels
may be used.
Ordinal The data exhibit the properties of nominal data, and the order or rank of the data is
meaningful. Note that ordinal data can also be recorded using a numeric code.
Interval The data show the properties of ordinal data, and the interval between values is expressed
in terms of a fixed unit of measure. Interval data are always numeric.
Ratio The data show the properties of interval data, and the ratio of two values is meaningful.
Variables such as distance, height, weight, and time use the ratio scale of measurement. This
scale requires that a zero value be included.
1
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