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Correlation and regression

Measuring linear association and fitting regression lines for prediction.

  • Define and explain Correlation and regression in your own words
  • Use key terms such as Binomial distribution accurately
  • Apply what you have learned to new examples and questions
  • Avoid the common mistakes learners make with this topic

This lesson focuses on Correlation and regression: measuring linear association and fitting regression lines for prediction.

Definition: Correlation and regression

Measuring linear association and fitting regression lines for prediction.

Key ideas

Binomial probabilities

P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ. The conditions matter: fixed n, two outcomes, constant p, independent trials. For n = 5, p = 0.3: P(X = 2) = 10 × 0.09 × 0.343 = 0.3087.

Standardising the normal

Convert X ~ N(μ, σ²) to Z ~ N(0, 1) via z = (x − μ)/σ, then read probabilities from tables. The total area under the curve is 1, and the curve is symmetric about μ, so the probability that X falls below the mean μ is 0.5.

Key term — Binomial distribution: The distribution of the number of successes in n independent trials each with success probability p, written X ~ B(n, p).

A binomial probability

X ~ B(5, 0.3). Find P(X = 2).

Use P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ with n = 5, r = 2, p = 0.3. C(5, 2) = 10; 0.3² = 0.09; 0.7³ = 0.343. Multiply: 10 × 0.09 × 0.343 = 0.3087.

Answer: P(X = 2) = 0.3087 (to 4 decimal places).

Common mistakes
  • Interpreting the regression line outside the data range Extrapolating beyond observed x-values assumes the linear relationship continues, which is often false.
  • Using the binomial model when trials are not independent Check the conditions: without replacement from a small population, p changes each trial and the binomial does not apply.

Practice

X ~ B(4, 0.5). Find P(X = 0).
All four trials fail.

P = C(4,0) × 0.5⁰ × 0.5⁴ = 0.0625.

X ~ B(6, 0.2). Find P(X = 1).
C(6,1) = 6.

P = 6 × 0.2 × 0.8⁵ = 6 × 0.2 × 0.32768 = 0.393216.

X ~ N(100, 15²). Find the z-score for x = 130.
z = (x − μ)/σ.

z = (130 − 100)/15 = 2.

For X ~ N(μ, σ²), what is P(X greater than μ)?
Think about symmetry.

0.5 — half the area lies above the mean.

Quick check

Correlation and regression — quick check

Which of these best defines "Binomial distribution"?

The distribution of the number of successes in n independent trials each with success probability p, written X ~ B(n, p).

Explain why 'the mean has increased' is a one-tailed alternative hypothesis.

It specifies an increase only, so the critical region lies in a single (upper) tail of the distribution.

A test at the 5% level gives p-value 0.032. What is the conclusion?

0.032 is less than 0.05, so reject H₀; there is sufficient evidence against the null hypothesis.
Key takeaways
  • Correlation and regression: measuring linear association and fitting regression lines for prediction.
  • Binomial probabilities: P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ.
  • Normal distribution: A continuous symmetric bell-shaped distribution, fully described by its mean μ and variance σ², written X ~ N(μ, σ²).
  • Watch out for: interpreting the regression line outside the data range
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