Enter the z-score to calculate the p-value. Select one-tailed or two-tailed and specify the significance level to determine if your result is statistically significant.

**P-Value:**

## Understanding Z-Scores, P-Values, and Hypothesis Testing

### 1. What is a Z-Score?

A Z-score (or standard score) tells us how many standard deviations an observation or data point is from the mean of a distribution. It's calculated using the formula:

where \( X \) is the observed value, \( \mu \) is the population mean, and \( \sigma \) is the population standard deviation.

### 2. Converting Z-Scores to P-Values

The P-value represents the probability of obtaining results as extreme as the observed results, assuming the null hypothesis is true. For a given Z-score, the P-value depends on the type of test:

**One-tailed test:**\( \text{P-value} = 1 - \Phi(|Z|) \) for an upper-tailed test, or \( \Phi(-|Z|) \) for a lower-tailed test, where \( \Phi \) is the cumulative distribution function (CDF) of the standard normal distribution.**Two-tailed test:**\( \text{P-value} = 2(1 - \Phi(|Z|)) \), which accounts for both tails of the distribution.

### 3. Cumulative Distribution Function (CDF) for the Standard Normal Distribution

The CDF of the standard normal distribution, \( \Phi(Z) \), gives the probability that a standard normal variable is less than or equal to a given Z-score. It represents the area under the normal curve up to the specified Z value.

To compute \( \Phi(Z) \), here are common approaches:

**Statistical Tables:**Use a Z-table to find the probability values for different Z-scores.**Software:**Most statistical software and libraries include a CDF function:**Python:**`scipy.stats.norm.cdf(z_score)`

**Excel:**`=NORM.S.DIST(z_score, TRUE)`

**JavaScript (jStat):**`jStat.normal.cdf(z_score, 0, 1)`

**Formula Approximation:**The CDF can also be approximated by:\( \Phi(Z) \approx 0.5 \left(1 + \text{erf}\left(\frac{Z}{\sqrt{2}}\right)\right) \)where`erf`

is the error function.

### 4. Interpreting the Results

In hypothesis testing, we compare the P-value to a significance level (α) to decide whether to reject the null hypothesis (H₀):

- If \( \text{P-value} < \alpha \): Reject the null hypothesis (suggesting the observed result is statistically significant).
- If \( \text{P-value} \geq \alpha \): Fail to reject the null hypothesis (the result is not statistically significant).

Common significance levels are 0.05 and 0.01.

### Real-Life Example: Manufacturing Quality Control

Imagine a factory producing screws with:

- Target length (\( \mu \)) = 5 cm
- Standard deviation (\( \sigma \)) = 0.1 cm
- Significance level (\( \alpha \)) = 0.05

#### One-Tailed Example (Upper-tail):

Testing if a 5.1 cm screw is significantly longer than the target:

**State hypotheses:**- H₀: μ = 5 cm
- H₁: μ > 5 cm

**Calculate Z-score:**

Z = (5.1 - 5) / 0.1 = 1.0**Find P-value:**

P-value = 1 - Φ(1.0) = 0.1587**Decision:**

Since 0.1587 > 0.05, fail to reject H₀

#### Two-Tailed Example:

Testing if a 5.2 cm screw significantly differs from the target:

**State hypotheses:**- H₀: μ = 5 cm
- H₁: μ ≠ 5 cm

**Calculate Z-score:**

Z = (5.2 - 5) / 0.1 = 2.0**Find P-value:**

P-value = 2(1 - Φ(2.0)) = 2(0.0228) = 0.0456**Decision:**

Since 0.0456 < 0.05, reject H₀

## Implementations

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.