ELEMENTI DI MATEMATICA E STATISTICA - FISICA

Module ELEMENTI DI MATEMATICA E STATISTICA

At the end of the course, students will be able to:

1. Basic mathematical skills

   • Apply fundamental concepts of mathematical analysis and statistics to model phenomena in pharmaceutical and biotechnological contexts, in line with Goal 4 – Quality education.

2. Data analysis skills

   • Use descriptive and inferential statistics to interpret experimental data, supporting research in health sciences, in coherence with Goal 3 – Good health and well-being.

3. Critical and quantitative thinking

   • Develop the ability to critically analyze experimental results, acknowledging uncertainty and variability, to foster robust and reliable scientific research, in relation to Goal 9 – Industry, innovation and infrastructure.

4. Sustainability and responsibility

   • Apply quantitative methods to assess and optimize production processes, contributing to efficient resource use and waste reduction, in line with Goal 12 – Responsible consumption and production.

5. Transferable skills and collaboration

   • Communicate quantitative data and results clearly and rigorously, including in interdisciplinary contexts, fostering international scientific cooperation, in coherence with Goal 17 – Partnerships for the goals.

In-person lectures
Lectures will be conducted in person, in accordance with current regulations.

According to the RDA, Art. 12 – University Educational Credits (CFU), within the standard workload of 25 hours of overall student commitment, corresponding to one credit, the following may apply:
a) 7 hours dedicated to lectures or equivalent teaching activities, with the remaining hours devoted to individual study;
b) at least 12 and no more than 15 hours dedicated to classroom exercises or equivalent assisted activities (laboratories), with the remaining hours devoted to personal study and review.

To successfully attend the course, students are expected to have prior knowledge of:

1. High-school level mathematics

   • Arithmetic: operations with real numbers, powers, roots, logarithms

   • Algebra: basic equations and inequalities, simple linear systems

   • Geometry: fundamental notions of analytic geometry in the plane (lines, circles)

   • Elementary functions: polynomial, exponential, logarithmic, and basic trigonometric functions

2. Logical skills

   • Ability to use deductive reasoning and solve simple quantitative problems

   • Understanding and interpreting graphs and tables

3. Basic computer literacy (recommended)

   • Elementary use of spreadsheets for calculations and graphical data representation.

The course provides the mathematical and statistical foundations required for data analysis and modeling in pharmaceutical and biotechnological contexts.

1. Preliminary concepts of mathematics

   • Sets, basic logic, and numerical operations

   • Proportions, percentages, growth rates

   • Radicals, powers, and logarithms

   • Basic equations and inequalities

2. Functions and differential calculus

   • Elementary functions: polynomial, exponential, logarithmic, trigonometric

   • Limits and continuity

   • Derivatives and applications: maxima, minima, function analysis

   • Approximations and Taylor expansion basics

3. Integral calculus (introduction)

   • Indefinite and definite integrals

   • Applications of integrals to areas and averages

4. Elements of linear algebra (introduction)

   • Vectors and matrices

   • Linear systems and methods of solution

5. Probability

   • Sample spaces, events, and probability

   • Conditional probability and independence

   • Random variables: discrete and continuous

   • Main distributions: binomial, Poisson, normal

   • Limit theorems (overview)

6. Statistics

   • Descriptive statistics: frequency tables, histograms, measures of central tendency and dispersion, boxplots

   • Bivariate statistics: correlation and linear regression

   • Inferential statistics:

     • Point and interval estimation

     • Hypothesis testing (Z-test, t-test, tests on proportions and variances)

     • Concepts of significance and p-value.

Exam dates are published on the website of the Department of Drug and Health Sciences:

https://www.dsf.unict.it/corsi/l-29_sfa/calendario-esami

Assessments may also be carried out online, if circumstances require it.

The exam consists of a written test and an oral test. Admission to the oral test is granted only upon passing the written test.

The evaluation will be assigned according to the following criteria:

Grade 29–30 with honors

• In-depth knowledge of the subject

• Ability to integrate and critically analyze the situations presented

• Independent resolution of complex problems

• Excellent communication skills and mastery of language

Grade 26–28

• Good knowledge of the subject

• Ability to perform clear and critical analysis of situations

• Fairly independent resolution of complex problems

• Clear presentation and appropriate language

Grade 22–25

• Fair knowledge, limited to the main topics

• Critical analysis not always consistent

• Fairly clear presentation with acceptable use of language

Grade 18–21

• Minimal knowledge of the subject

• Limited ability to integrate and critically analyze situations

• Sufficiently clear presentation, with weak language proficiency

Exam not passed

• Lack of knowledge of the main contents

• Very limited or no ability to use specific terminology

• Inability to independently apply acquired knowledge

Compensatory and dispensatory measures

To make sure everyone has the same opportunities and in line with current laws, students who need support can ask for a personal meeting to plan possible compensatory or exemption measures, according to the course goals and their specific needs.

You can also contact the CInAP representative of our Department (Center for Active and Participatory Integration – Services for Disabilities and/or Specific Learning Disorders), Prof. Santina Chiechio.

Basic Mathematics

1. Solve the equation $\log (x^2-3)=1$.

2. Determine the domain and monotonicity of the function $f(x)=\frac{x^2-1}{x-2}$.

Differential and Integral Calculus
3. Compute ${\lim }_{x\to 0}\frac{\sin (3x)}{x}$.
4. Find the maxima and minima of the function $f(x)=x^3-3x^2+2$.
5. Compute the area enclosed between the curve $y=x^2$ and the line $y=2x$.

Linear Algebra
6. Solve the linear system:

Probability
7. A biased coin has probability $p=0.6$ of landing heads. What is the probability of obtaining exactly 3 heads in 5 tosses?
8. Two events $A$ and $B$ are independent with $P(A)=0.4$ and $P(B)=0.5$. Compute $P(A\cup B)$.

Statistics
9. A sample of 50 patients has an average cholesterol level of 200 mg/dl with a standard deviation of 20. Construct a 95% confidence interval for the population mean.
10. In a clinical trial, out of 100 patients, 60 showed improvement after treatment. Test at the 0.05 significance level whether the probability of success is greater than 50%.