In Mathematics, especially in the topic of Hypothesis Tests, a deep understanding of Type I and Type II errors is crucial. These concepts are not just mathematical abstractions but have real-world implications, affecting how we interpret data and make decisions based on statistical analysis.

**Understanding Hypothesis Testing Errors:**

Hypothesis testing helps determine if sample data reflects a population condition. However, errors can lead to wrong conclusions.

**Type I Error (False Positive):**

**What is it?**Rejecting a true null hypothesis.**Example:**Declaring a new drug effective when it's not.**Statistical Significance (**$\alpha$**):**The chance of a Type I error, typically set at 0.05, 0.01, or 0.10.**Choosing (**$\alpha$**):**Depends on the stakes of making a Type I error; lower (\alpha) for higher stakes.

**Type II Error (False Negative):**

**What is it?**Accepting a false null hypothesis.**Example:**Missing the effectiveness of a good drug.**Represented by (**$\beta$**):**The probability of a Type II error.**Power of a Test:**The likelihood of correctly rejecting a false null hypothesis $1 - (\beta)$. Increase power by enlarging sample size, adjusting $\alpha$, or improving design.

**Balancing Errors:**

**The Challenge:**Lowering one error type raises the other.**Risk Management:**The balance between Type I and II errors depends on their consequences. Choose based on what's at stake.

## Example Problem: Quality Control Hypothesis Test

### Problem:

A machine is supposed to make screws of a specific diameter. Quality control tests if it does correctly.

- Null Hypothesis $(H_0)$: Mean diameter is as specified.
- Alternative Hypothesis $(H_1)$: Mean diameter differs.
- Significance Level $(\alpha)$ = 0.05
- Sample Size $(n)$ = 30

### Calculation:

- Assume sample mean $(\bar{x})$ = 8.02 mm, population mean ((\mu)) = 8 mm, standard deviation $(\alpha)$ = 0.2 mm.
- Standard Error (SE): $SE = \frac{\sigma}{\sqrt{n}}$
- Z-Score: $Z = \frac{\bar{x} - \mu}{SE}$
- Decision: If |Z| > Z_{\alpha/2}, reject $H_0$.

### Solution:

Given $\bar{x} = 8.02$ mm, $\mu = 8$ mm, $\sigma = 0.2$ mm, and $n = 30$:

SE = $\frac{0.2}{\sqrt{30}}$

Z = $\frac{8.02 - 8}{SE}$

Critical Z-Value = 1.96 (for $\alpha = 0.05$)

Since the calculated Z is less than 1.96, we do not reject $H_0$. There's insufficient evidence to say the machine is incorrect.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.