### 1. Algebra:

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. Here are some fundamental algebraic concepts:

**Variables and Expressions: **Variables represent unknown or changing quantities. Expressions are combinations of variables, numbers, and operations (addition, subtraction, multiplication, division).

a + b (addition of two numbers)

a − b (subtraction of two numbers)

a × b or a * b (multiplication of two numbers)

a / b (division of two numbers)

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**Equations and Inequalities: **Equations express the equality of two expressions. Inequalities express relationships where one side is greater than or less than the other.

$ax+b=0$ (linear equation)

$ax+bx+c=0$ (quadratic equation)

$a>b$ (inequality)

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**Functions: **Functions describe relationships between variables. They have inputs (independent variable) and outputs (dependent variable).

$f(x)=ax+b$ (linear function)

$f(x)=ax+bx+c$ (quadratic function)

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### 2. Calculus:

Calculus involves the study of rates of change and the accumulation of quantities. Key concepts include:

**Limits and Continuity: **Limits describe the behavior of a function as the input approaches a certain value. Continuity ensures that there are no abrupt jumps or holes in the graph of a function.

$limx→a f(x)=L$ (limit of a function)

$f(x)$ is continuous at $x=a$

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**Derivatives: **Derivatives measure the rate at which a function changes. They represent slopes of tangent lines to curves.

$_{f′}(x)$ or $df/dx $ (derivative of $f(x)$)

$d/dx [ax]=nax$^{ }(power rule)

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**Integrals: **Integrals compute the accumulation of quantities and the area under curves. They are the reverse process of derivatives.

$∫f(x)dx$ (indefinite integral)

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### 3. Statistics:

Statistics involves the collection, analysis, interpretation, presentation, and organization of data. Key concepts include:

**Descriptive Statistics: **Measures of central tendency (mean, median, mode). Measures of dispersion (range, variance, standard deviation).

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**Inferential Statistics: **Uses sample data to make inferences about a population. Includes hypothesis testing and confidence intervals.

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### 4. Probability:

Probability deals with the likelihood of events occurring. Key concepts include:

**Probability Basics: **Probability is a number between 0 and 1, representing the likelihood of an event. The sum of all probabilities in a sample space is 1.

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**Random Variables and Probability Distributions: **Random variables represent outcomes of a random process. Probability distributions describe the likelihood of different outcomes.

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### Hypothesis Testing:

Hypothesis testing is a statistical method to make inferences about a population based on a sample of data. Key steps include:

**Formulating Hypotheses: **Null hypothesis (H0) and alternative hypothesis (H1).

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**Choosing a Significance Level: **Common levels include 0.05 or 0.01.

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**Collecting Data and Calculating Test Statistic: **Use statistical tests (t-tests, chi-square tests) to analyze data.

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**Making a Decision: **Compare the p-value to the significance level and make a decision about the null hypothesis.

Compare the p-value to the significance level (α)

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Compare the p-value to ($α)$ to accept or reject the null hypothesis.