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.