Statistics What is Null and Alternate Hypothesis?

In statistics, we often want to test a claim or theory about a population (e.g., "This new drug is effective," "Our new website design increases sales"). Since we can't test the entire population, we use sample data. The Null and Alternative Hypotheses are two competing, mutually exclusive statements about this population.

Of course. This is a fundamental concept in statistics used in hypothesis testing.

### The Core Idea

In statistics, we often want to test a claim or theory about a **population** (e.g., "This new drug is effective," "Our new website design increases sales"). Since we can't test the entire population, we use sample data. The **Null** and **Alternative Hypotheses** are two competing, mutually exclusive statements about this population.

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### 1. The Null Hypothesis (\(H_0\))

*   **What it is:** The **default or status quo** assumption. It's a statement of "no effect," "no difference," or "no change." It represents skepticism.

*   **Symbol:** \(H_0\)

*   **It always contains an equality:** \(=\), \(\leq\), or \(\geq\).

*   **The goal of a hypothesis test is to gather evidence *against* the null hypothesis.**

    **Examples:**

    *   A new drug is no better than a placebo. (\(H_0: \mu_{\text{drug}} = \mu_{\text{placebo}}\))

    *   The mean height of men is 175 cm. (\(H_0: \mu = 175\))

    *   The proportion of defective items is less than or equal to 2%. (\(H_0: p \leq 0.02\))

Think of it like a courtroom principle: **The defendant is innocent until proven guilty.** The null hypothesis is the assumption of innocence.

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### 2. The Alternative Hypothesis (\(H_1\) or \(H_a\))

*   **What it is:** The **researcher's claim** or what you hope to prove. It's a statement that contradicts the null hypothesis. It represents a new effect, difference, or change.

*   **Symbol:** \(H_1\) or \(H_a\)

*   **It never contains an equality:** \(\neq\), \(>\), or \(<\).

*   **We only accept the alternative hypothesis if the sample data provides strong enough evidence to *reject* the null hypothesis.**

    **Examples (corresponding to the nulls above):**

    *   The new drug is better than the placebo. (\(H_a: \mu_{\text{drug}} > \mu_{\text{placebo}}\))

    *   The mean height of men is not 175 cm. (\(H_a: \mu \neq 175\))

    *   The proportion of defective items is greater than 2%. (\(H_a: p > 0.02\))

In the courtroom analogy, this is the **prosecution's claim** that the defendant is guilty.

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### How They Work Together

1.  **State the Hypotheses:** You define both \(H_0\) and \(H_a\) before collecting data.

2.  **Collect Sample Data:** You gather evidence from the real world.

3.  **Perform a Statistical Test:** This calculates a probability (p-value) of observing your sample data *if the null hypothesis were true*.

4.  **Make a Decision:**

    *   If the evidence is very unlikely under \(H_0\) (p-value is low), you **reject the null hypothesis** in favor of the alternative. This is like finding the defendant "guilty."

    *   If the evidence is not unlikely under \(H_0\) (p-value is high), you **fail to reject the null hypothesis.** This is like a verdict of "not guilty." (Note: We never "accept" the null; we just don't have enough evidence to reject it).

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### Key Takeaway Table

| Feature | Null Hypothesis (\(H_0\)) | Alternative Hypothesis (\(H_a\)) |

| :--- | :--- | :--- |

| **Represents** | Status quo, no effect, no difference | Researcher's claim, an effect, a difference |

| **Symbol** | \(H_0\) | \(H_1\) or \(H_a\) |

| **Contains** | \(=\), \(\leq\), \(\geq\) | \(\neq\), \(>\), \(<\) |

| **Court Analogy** | Innocence | Guilt |

| **Goal of Test** | Gather evidence to **reject** it | Gather evidence to **support** it |

**Analogy Summary:** You assume the null hypothesis is true (like assuming innocence). The sample data is the evidence. If the evidence is strong enough against the null, you reject it and side with the alternative.