What is two tailed Hypothesis test. When it is used

   

## Explanation:

In a **two-tailed hypothesis test**, the rejection region is **split between both tails** of the distribution.

## Visual Representation:

```

Two-Tailed Test (α = 0.05)

Rejection Region: Both tails (2.5% in each tail)

         │

    ┌────┼────┐

    │    │    │

    │    │    │

    │    │    │

[####]   │   [####]   ← Rejection regions (2.5% each)

    │    │    │

    │    │    │

    │    │    │

-1.96   0   1.96     ← Critical values

```

## Mathematical Confirmation:

```python

from scipy import stats

# For α = 0.05 two-tailed test:

alpha = 0.05

critical_value = stats.norm.ppf(1 - alpha/2)  # 1.96

print(f"Two-tailed critical values: ±{critical_value:.3f}")

print(f"Rejection region: z < -{critical_value:.3f} OR z > {critical_value:.3f}")

print(f"Area in left tail: {alpha/2:.3f} ({alpha/2*100}%)")

print(f"Area in right tail: {alpha/2:.3f} ({alpha/2*100}%)")

```

**Output:**

```

Two-tailed critical values: ±1.960

Rejection region: z < -1.960 OR z > 1.960

Area in left tail: 0.025 (2.5%)

Area in right tail: 0.025 (2.5%)

```

## Why This is True:

### **Two-Tailed Test Logic:**

- **H₀:** μ = μ₀ (No difference)

- **H₁:** μ ≠ μ₀ (Difference in EITHER direction)

- We reject H₀ if the test statistic is **significantly large OR significantly small**

- Therefore, we need **rejection regions on both sides**

### **Comparison with One-Tailed Tests:**

| Test Type | Rejection Region | Hypothesis |

|-----------|------------------|------------|

| **Two-Tailed** | **Both tails** | H₁: μ ≠ μ₀ |

| **Right-Tailed** | Right tail only | H₁: μ > μ₀ |

| **Left-Tailed** | Left tail only | H₁: μ < μ₀ |

## Medical Example:

```python

# Testing if a drug changes blood pressure (could increase OR decrease)

# Two-tailed test is appropriate

print("Two-tailed test scenario:")

print("H₀: Drug has NO effect on blood pressure (μ = 120)")

print("H₁: Drug CHANGES blood pressure (μ ≠ 120)")

print("→ We reject if blood pressure is significantly HIGHER OR LOWER")

print("→ Therefore, rejection regions on BOTH sides")

```

## Key Point:

The statement **"In a two-tailed hypothesis test, the rejection region lies on both sides of the distribution"** is **definitely TRUE** and represents the fundamental characteristic that distinguishes two-tailed tests from one-tailed tests.