Inferential Statistics

What is Inferential Statistics? #

Inferential statistics is used to make conclusions about a population based on a sample.

Example:
Survey 100 people → predict behavior of 1 million

Probability Basics #

What is Probability? #

Probability measures how likely an event is to happen$$0 \leq P(Event) \leq 1$$0≤P(Event)≤1

• 0 → Impossible
• 1 → Certain

Formula #

$$P(E) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}$$P(E)=Total outcomesFavorable outcomes​

Example #

Toss a coin$$P(Heads) = \frac{1}{2} = 0.5$$P(Heads)=21​=0.5

Key Concepts #

Independent Events #

• One event doesn’t affect another
  xample: coin tosses

Conditional Probability #

• Probability depends on another event

$$P(A|B)$$P(A∣B)

Normal Distribution #

What is Normal Distribution? #

A bell-shaped curve where:

• Most values are near the mean
• Symmetrical distribution

Key Properties #

• Mean = Median = Mode
• Symmetrical
• Defined by:
  • Mean (μ)
  • Standard Deviation (σ)

Empirical Rule (Important) #

Example #

Exam scores:

• Mean = 70
• Std Dev = 10
• 68% students → between 60–80

Z-Score #

Definition #

Z-score tells how far a value is from the mean (in standard deviations)

Formula #

$$Z = \frac{X – \mu}{\sigma}$$Z=σX−μ​

Example #

• X = 80
• Mean = 70
• Std Dev = 10

$$Z = (80 – 70) / 10 = 1$$Z=(80−70)/10=1

Value is 1 standard deviation above mean

Use #

• Detect outliers
• Standardize data
• Compare different datasets

Hypothesis Testing #

What is Hypothesis Testing? #

A method to test assumptions using data

Key Terms #

Example #

• H₀: New drug has no effect
• H₁: New drug works

p-value #

Definition #

p-value = probability of getting results if H₀ is true

Decision Rule #

Meaning #

• Small p-value → strong evidence
• Large p-value → weak evidence

T-Test #

What is T-Test? #

Used to compare means of two groups

Types #

Example #

Compare:

• Old method vs New method scores

Chi-Square Test #

What is Chi-Square? #

Used for categorical data to test relationships

Example #

Test:

• Gender vs Product preference

Use #

• Check independence
• Compare observed vs expected

Correlation vs Causation #

Correlation #

Measures relationship between variables

• Positive → both increase
• Negative → one increases, other decreases

Example #

• Ice cream sales ↑
• Temperature ↑

Correlated

9.3 Causation #

One variable directly causes another

Key Difference #

Important Note #

Correlation ≠ Causation

Summary Table #