Hypothesis Testing

👋, readers!

Here will introduce you to some fundamentals of probability theory. These will help grasp the important concepts of hypothesis testing.

  1. Probability Visualization

  2. Assigning Probabilities

  3. General Probability Rules

  4. Probability to Hypothesis Testing

Also, you will take some quizzes and interact with some visualizations.

Probability Visualization

Accessible Statistics

Seeing Theory created by Daniel Kunin while at Brown University wanted to make the concept of probability and statistics easier to understand. Feel free to see through all of its content. But for this presentation, look at Basic Probability.

Seeing Theory

Do you want to go to that website? See here

Assigning Probabilities

In some point, you will face the situation where you need to assign probabilities. So, here will show you to assign probabilities.

Let’s do something simple. How many possibilities does one coin have? The answer is 2, so the formula for sample space is: S = {Head, Tail}. This means 1/2.

How many possibilities does one dice have? The answer is 6, so the formula for sample space is: S = {1, 2, 3, 4, 5, 6}. This means 1/6.

How many possibilities do two coins have? The answer is 4, so the formula for sample space is: S = {(H, H), (H, T), (T, H), (T, T)}. This means 1/4.

How many possibilities do two dices have? This question becomes little harder. To solve this, you can use some math. The formula is: Total Possibilities = (n1)(n2)…(nk). For example, the first one is coin, which has 2 possibilities. The second one is dice, which has 6 possibilities. Then 2 x 6 = 12. What is your answer for two dices?

The correct answer is 36. This means 1/36.

General Probability Rules

Rule 1:

Range of possible probabilities: 0 ≤ P(A) ≤ 1

Rule 2:

Sample space of all possibilities: P(S) = 1

Example: If probability of deaf students is P(d) and probability of hearing students P(h), then P(d) + P(h) = 1.

Rule 3:

For any event, P(Ac) = 1 - P(A).

Example: If you are NOT registered nurses, then P(nc) = 1 - P(n).

Rule 4a - Addition Rule:

If two events are mutually exclusive, then:

P(A ∪ B) = P(A) + P(B)

Rule 4b - Addition Rule:

If two event are NOT mutually exclusive, then:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Rule 5 - Multiplication Rule:

5a: If A and B events are independent, then:

P(A ∩ B) = P(A) • P(B)

5b: If A and B events somehow affect each other, then:

P(A ∩ B) = P(A) • P(B|A)

Example of Rule 5b

The bag contains 5 red marbles and 10 purple marbles. When you draw the one marble, you remove this marble out of the bag. This process is called without replacement. If you draw the first red marble, the probability will be:

P(A) = P(first red marble) = 5/15

The second draw is when you put condition on probability. That is, the conditional probability of B on A is:

P(B|A) = 4/14

Thus:

P(A ∩ B) = P(A) • P(B|A) = 5/15 • 4/14

Rule 6 - Conditional Probab.:

P(A | B) = P(A ∩ B) • P(A)-1

Example:

P(Biomedical Students | Deaf) = 15/55 ≈ 0.27

P(Deaf | Biomedical Students) = 15/65 ≈ 0.23

Summary:

  • Rule 1 - Rule 3 is basic principle of probabilities.

  • Addition Rule is in use when you want to know that the probability of at least one of the events occurs.

  • Multiplication Rule is in use when you want to know that probability of both events occurs.

  • Conditional Probability Rule is in use when due to another event, you want to know how likely the event occurs.

Exercises

Probability to Hypothesis Testing