The Normal Distribution and Hypothesis Testing

The normal distribution

The normal distribution is a continuous, bell-shaped, symmetrical distribution, written X ~ N(μ, σ²) where μ is the mean and σ the standard deviation. Features:

• Symmetrical about the mean; mean = median = mode.
• About 68% of data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.
• The total area under the curve = 1; probabilities are areas under the curve.

Standardising

To use tables/calculator, convert to the standard normal Z ~ N(0, 1):

Z = (X − μ) ÷ σ

Z tells you how many standard deviations a value is from the mean. Use P(Z < z) values (from a calculator or tables).

Normal approximation to the binomial

If n is large and p is not too extreme (np and n(1−p) both > 5), B(n, p) can be approximated by N(np, np(1−p)) — with a continuity correction.

Hypothesis testing

A hypothesis test uses sample data to decide between two claims about a population parameter.

• Null hypothesis (H₀): the "no change" statement (e.g. p = 0.5).
• Alternative hypothesis (H₁): what you're testing for (e.g. p > 0.5 one-tailed, or p ≠ 0.5 two-tailed).
• Significance level (e.g. 5%) is the probability of wrongly rejecting H₀.

Method:

1. State H₀ and H₁ and the significance level.

2. Assume H₀ is true and model the situation (e.g. binomial or normal).

3. Calculate the probability of the observed (or more extreme) result — the p-value — or find the critical region.

4. If the result lies in the critical region (p-value < significance level), reject H₀; otherwise there is insufficient evidence to reject it.

5. Write the conclusion in context.

For a two-tailed test, split the significance level between the two tails.

Worked example

A coin is flipped 20 times, landing heads 15 times. Test at 5% whether it is biased towards heads (H₀: p = 0.5, H₁: p > 0.5).

• Under H₀, X ~ B(20, 0.5). P(X ≥ 15) = 1 − P(X ≤ 14) ≈ 0.021.
• 0.021 < 0.05 → reject H₀: there is evidence the coin is biased towards heads. ✓

Common mistakes

• Mixing up μ and σ, or forgetting σ² is the variance in N(μ, σ²).
• Getting the standardising formula upside down.
• Not stating hypotheses or the conclusion in context; using one tail when the test is two-tailed.

Exam tips

• Always define H₀ and H₁ clearly and state the significance level.
• Compare the p-value to the significance level (or check the critical region).
• Conclude in context ("there is/insufficient evidence that…").

Key facts to remember

• Normal N(μ, σ²): symmetrical bell curve; standardise with Z = (X − μ)/σ; ~68/95/99.7% within 1/2/3σ.
• Large-n binomial ≈ N(np, np(1−p)).
• Hypothesis test: state H₀/H₁ + significance level, assume H₀, find p-value/critical region, reject if in the critical region, conclude in context.