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CIE A-Level Maths Study Notes

5.5.3 Hypothesis Testing for Population Means

Hypothesis testing is a statistical method used to infer whether a hypothesis about the population mean is true or not. It's particularly crucial in cases where populations are normally distributed and when dealing with large sample sizes.

Introduction to Hypothesis Testing

In hypothesis testing, we start by assuming a null hypothesis (H₀), typically a statement indicating no effect or no change. This hypothesis is tested against an alternative hypothesis (H₁), which represents a significant effect or change.

Key Elements of Hypothesis Testing:

  • Hypothesis Testing: Determines if a claim about a population mean is true.
  • Null Hypothesis (H0H_0): No change or effect. Example: "Population mean is X."
  • Alternative Hypothesis (H1H_1): Indicates change or effect. Example: "Population mean is not X."
  • Test Statistic: Number from sample data to test H₀.
  • Significance Level (α): Usually 5% (0.05), marks statistical significance.
  • Rejection Region: Test statistic values that lead to rejecting H₀.
  • Acceptance Region: Values where H0H_0 is not rejected.

Scenarios

1. Known Population Variance: Use normal distribution for test statistic.

2. Large Samples (n ≥ 30): Normal approximation is valid due to Central Limit Theorem.

Hypothesis Testing Steps

1. State Hypotheses: Define H0H_0 and H1H_1.

2. Choose α: Commonly 0.05.

3. Calculate Test Statistic: Use formula based on variance and sample size.

4. Determine Rejection Region: Based on α and test statistic distribution.

5. Decision: Reject or not reject H₀ based on calculations.

Example

A bakery claims that the average weight of their bread loaves is 500 grams. To test this claim, a random sample of 36 loaves is selected, and the average weight is found to be 505 grams with a standard deviation of 15 grams. Test the bakery's claim at a 5% significance level.

Solution:

1. Hypotheses

  • H₀: μ = 500 grams
  • H₁: μ ≠ 500 grams

2. Significance Level

  • α = 0.05

3. Test Statistic Calculation

  • Sample mean (xˉ)(\bar{x}) = 505 grams
  • Population mean under H₀ (μ) = 500 grams
  • Standard deviation (σ) = 15 grams
  • Sample size nn
  • = 36
  • Z=50550015/36=52.5Z = \frac{505 - 500}{15/\sqrt{36}} = \frac{5}{2.5} = 2.00

4. Rejection Region

  • Critical Z-value ≈ ±1.96
  • Rejection if |Z| > 1.96

5. Decision

  • Calculated Z = 2.00 > 1.96
  • Reject H₀

6. Conclusion

Evidence suggests average weight is not 500 grams.

Normal Distribution 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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