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CIE A-Level Maths Study Notes

5.5.4 Understanding Type I and Type II Errors

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 (α\alpha): The chance of a Type I error, typically set at 0.05, 0.01, or 0.10.
  • Choosing (α\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 (β\beta): The probability of a Type II error.
  • Power of a Test: The likelihood of correctly rejecting a false null hypothesis 1(β)1 - (\beta). Increase power by enlarging sample size, adjusting α\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 (H0)(H_0): Mean diameter is as specified.
  • Alternative Hypothesis (H1)(H_1): Mean diameter differs.
  • Significance Level (α)(\alpha) = 0.05
  • Sample Size (n)(n) = 30

Calculation:

  • Assume sample mean (xˉ)(\bar{x}) = 8.02 mm, population mean ((\mu)) = 8 mm, standard deviation (α)(\alpha) = 0.2 mm.
  • Standard Error (SE): SE=σnSE = \frac{\sigma}{\sqrt{n}}
  • Z-Score: Z=xˉμSEZ = \frac{\bar{x} - \mu}{SE}
  • Decision: If |Z| > Z_{\alpha/2}, reject H0H_0.

Solution:

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

SE = 0.230\frac{0.2}{\sqrt{30}}

Z = 8.028SE\frac{8.02 - 8}{SE}

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

Since the calculated Z is less than 1.96, we do not reject H0H_0. There's insufficient evidence to say the machine is incorrect.

Quality Control Hypothesis Test Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

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.

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