Statistical modelling is a fundamental technique in applied statistics, enabling professionals to make predictions, understand relationships, and provide evidence-based recommendations in fields such as banking, healthcare, government policy, and research. In Kenya, statistical models are essential for analyzing survey data, monitoring disease outbreaks, forecasting economic trends, and optimizing resource allocation. This chapter focuses on implementing statistical models, emphasizing calculation, interpretation, and presentation using real-world Kenyan contexts.

3.1 Statistical Modelling Model

Statistical modelling involves representing real-world data using mathematical equations to describe relationships between variables. Models such as regression, correlation, and probability distributions are central tools for applied statisticians working in sectors like finance, agriculture, and public health. The process includes formulating the model, estimating parameters, checking validity, and using the results for decision-making.

3.1.1 Definition of Terms

Statistical modelling uses specific terminology to describe data, relationships, and calculations. Understanding these terms is crucial for interpreting results and communicating findings.

Measures of Central Tendency and Dispersion

Worked Examples

1. Example 1: Calculating Mean (Easy)
   Given: Number of patients attended in a hospital over 5 days: 42, 38, 40, 45, 35$$ \text{Mean} = \frac{\Sigma x_{i}}{n} $$$$ \text{Mean} = \frac{42 + 38 + 40 + 45 + 35}{5} $$$$ \text{Mean} = \frac{200}{5} $$ Answer: 40 patients

2. Example 2: Finding Mode (Medium)
   Given: Sales (Ksh) at a retail shop over 7 days: 5200, 5400, 5200, 5600, 5200, 5800, 5400$$ \text{Mode} = \text{Value that appears most frequently} $$$$ \text{5200 appears 3 times, 5400 appears 2 times} $$ Answer: Mode = 5200 Ksh

3. Example 3: Calculating Variance (Hard)
   Given: Exam scores at a university: 60, 65, 70, 75, 80$$ \text{Variance} = \frac{\Sigma (x_{i} - \mu)^2}{n} $$$$ \mu = \frac{60 + 65 + 70 + 75 + 80}{5} = 70 $$$$ \text{Variance} = \frac{(60-70)^2 + (65-70)^2 + (70-70)^2 + (75-70)^2 + (80-70)^2}{5} $$$$ = \frac{100 + 25 + 0 + 25 + 100}{5} $$$$ = \frac{250}{5} $$ Answer: 50

4. Example 4: Calculating Standard Deviation (Applied)
   Given: Daily rainfall (mm) on a farm: 10, 12, 8, 15, 9$$ \text{Standard Deviation} = \sqrt{\text{Variance}} $$$$ \mu = \frac{10 + 12 + 8 + 15 + 9}{5} = 10.8 $$$$ \text{Variance} = \frac{(10-10.8)^2 + (12-10.8)^2 + (8-10.8)^2 + (15-10.8)^2 + (9-10.8)^2}{5} $$$$ = \frac{0.64 + 1.44 + 7.84 + 17.64 + 3.24}{5} $$$$ = \frac{30.8}{5} $$$$ = 6.16 $$$$ \text{Standard Deviation} = \sqrt{6.16} \approx 2.48 $$ Answer: 2.48 mm

5. Example 5: Calculating Range (Challenge)
   Given: Monthly salaries (Ksh): 32,000; 41,000; 29,000; 39,000; 36,000$$ \text{Range} = \text{Maximum} - \text{Minimum} $$$$ \text{Range} = 41,000 - 29,000 $$$$ \text{Range} = 12,000 $$ Answer: 12,000 Ksh

Practice Questions

3.1.2 Mathematical Symbols Used in Statistical Models

Statistical models employ mathematical symbols to concisely represent relationships and calculations. Greek letters such as \( \Sigma \), \( \mu \), \( \sigma \), \( \beta \), and \( \pi \) are commonly used in formulas for summarizing and analysing data.

Worked Examples

1. Example 1: Using Summation Symbol (Easy)
   Given: Values for environmental compliance scores: 12, 15, 18$$ \Sigma x_{i} = 12 + 15 + 18 $$$$ \Sigma x_{i} = 45 $$ Answer: 45

2. Example 2: Population Mean Symbol (Medium)
   Given: Population mean income for a town is Ksh 27,500. Write using symbol.$$ \mu = 27,500 $$ Answer: \(\mu = 27,500\) Ksh

3. Example 3: Standard Deviation Symbol (Hard)
   Given: Population standard deviation for test scores is 4.5.$$ \sigma = 4.5 $$ Answer: \(\sigma = 4.5\)

4. Example 4: Beta Symbol in Regression (Applied)
   Given: Regression model coefficient for hours studied is 2.3.$$ \beta = 2.3 $$ Answer: \(\beta = 2.3\)

5. Example 5: Product Symbol (Challenge)
   Given: Multiply values 5, 6, 7.$$ \pi = 5 \times 6 \times 7 $$$$ \pi = 210 $$ Answer: 210

Practice Questions

3.1.3 Extracting and Interpreting Information from Statistical Models

Extracting information from statistical models involves using tables, graphs, and summary statistics to draw conclusions and support decision-making. Applied statisticians interpret these outputs to guide actions in fields like health, finance, and education.

Worked Examples

1. Example 1: Reading Mean from a Table (Easy)
   Given: Table showing average daily sales in a shop as 3400, 3600, 3200, 3500, 3700$$ \text{Mean} = \frac{3400 + 3600 + 3200 + 3500 + 3700}{5} $$$$ = \frac{17,400}{5} $$$$ = 3,480 $$ Answer: 3,480 Ksh

2. Example 2: Interpreting Graph for Mode (Medium)
   Given: Bar graph of student attendance: 40, 55, 55, 50, 45$$ \text{Mode} = 55 $$ Answer: Mode = 55 students

3. Example 3: Variance from Summary Statistics (Hard)
   Given: Summary statistics for test scores: Mean = 65, Variance = 9. Find standard deviation.$$ \text{Standard Deviation} = \sqrt{9} $$$$ = 3 $$ Answer: 3

4. Example 4: Range from Table (Applied)
   Given: Table of monthly expenses (Ksh): 12,000; 14,500; 13,000; 15,000; 11,500$$ \text{Range} = 15,000 - 11,500 $$$$ = 3,500 $$ Answer: 3,500 Ksh

5. Example 5: Interpreting Quartiles from Box Plot (Challenge)
   Given: Box plot shows Q1 = 22, Q2 = 35, Q3 = 44 for age of patients.$$ \text{Interquartile Range} = Q3 - Q1 $$$$ = 44 - 22 $$$$ = 22 $$ Answer: 22 years

Practice Questions

3.1.4 Presenting Results Using Mathematical Language

Presenting statistical results effectively requires the use of mathematical language and symbols to ensure clarity and precision. This is especially important when communicating findings to stakeholders in sectors such as banking, hospitals, and government agencies.

Worked Examples

1. Example 1: Reporting Mean and Standard Deviation (Easy)
   Given: Mean income = 25,000 Ksh, Standard deviation = 4,000 Ksh$$ \mu = 25,000, \sigma = 4,000 $$ Answer: \(\mu = 25,000\) Ksh, \(\sigma = 4,000\) Ksh

2. Example 2: Using Mathematical Symbols in Summary (Medium)
   Given: Summation of scores (\( \Sigma x_{i} \)) = 340, Number of students (\( n \)) = 10$$ \text{Mean} = \frac{\Sigma x_{i}}{n} = \frac{340}{10} = 34 $$ Answer: Mean = 34

3. Example 3: Expressing Regression Equation (Hard)
   Given: Regression analysis in a bank yields \( y = 2.5x + 3 \)$$ \text{Where} \ y = \text{expected revenue}, \ x = \text{number of clients} $$ Answer: \( y = 2.5x + 3 \)

4. Example 4: Describing Standard Deviation and Variance (Applied)
   Given: Variance (\( \sigma^2 \)) = 16, Standard deviation (\( \sigma \)) = 4$$ \sigma^2 = 16, \sigma = 4 $$ Answer: Variance = 16, Standard deviation = 4

5. Example 5: Using Mathematical Symbols in Probability (Challenge)
   Given: Probability of event (\( P \)) = 0.75$$ P = 0.75 $$ Answer: Probability = 0.75

Practice Questions

Chapter Summary

This chapter explored the concept of statistical modelling, beginning with key definitions and the foundational theory behind modelling in data analysis. It explained the distinction between independent and dependent variables, and outlined different types of variables commonly used in statistical models. Practical aspects of statistical modelling were discussed, including hands-on exercises, illustrative examples, and the use of simulations to deepen understanding. The chapter then examined various statistical models, covering simple and multiple regression, non-linear models, and logistic regression, highlighting their characteristics and applications. Finally, guidance was provided on how to choose the most appropriate statistical model for a given dataset or research question. Together, these topics equip learners with a comprehensive toolkit for implementing effective statistical models in real-world scenarios.