H2 Maths Notes (JC 1-2): 6.3) Normal Distribution

Before you revise\ Refresh the discrete distribution rules (binomial, Poisson) so you know when normal modelling is appropriate. Bookmark the GC normal menu (normalcdf, invNorm) and keep a sketch pad handy—exam scripts still expect hand-drawn bell curves with shaded regions.

Core Concepts

• A continuous random variable \( X \) has the normal distribution \( X \sim \mathcal{N}(\mu, \sigma^2) \) if its density is bell-shaped, centred at \( \mu \), with spread controlled by \( \sigma \).
• Standardisation converts \( X \) to \( Z = \frac{X - \mu}{\sigma} \sim \mathcal{N}(0, 1) \); tables and GC functions assume this form.
• Symmetry gives \( P(Z \leq -a) = P(Z \geq a) \) and \( P(|Z| \leq a) = 2P(Z \leq a) - 1 \).
• Areas under the density curve equal probabilities because the total area is 1.
• Continuity correction improves approximations when a discrete distribution (e.g. binomial) is approximated by a normal: replace \( P\( X = k \) \) with \( P(k - 0.5 \leq X \leq k + 0.5) \).

Standardisation Workflow

1. Sketch the bell curve, mark \( \mu \) and shade the requested region.
2. Convert bounds using \( z = \frac{x - \mu}{\sigma} \).
3. Evaluate the corresponding standard normal probability with the table or GC.
4. State the final probability to three significant figures unless otherwise required.

Example -- Tail probability

Industrial bearings have \( X \sim \mathcal{N}(45.0, 1.2^2) \) mm. Bearings longer than \( 46.5 \) mm are scrapped. Find the scrap rate.

1. Compute \( z = \frac{46.5 - 45.0}{1.2} = 1.25 \).
2. \( P(X \geq 46.5) = P(Z \geq 1.25) = 1 - \Phi(1.25) \approx 0.106 \).
3. About 10.6% exceed the upper tolerance.

Continuity Correction and Approximations

When \( X \sim \operatorname{Bin}(n, p) \) with both \( np \geq 5 \) and \( n(1 - p) \geq 5 \), approximate by \( Y \sim \mathcal{N}(np, np(1 - p)) \).

Example -- Binomial to normal

An IP class sits 120-question quizzes with \( p = 0.6 \) success probability per item. Approximate the probability of scoring at least 80 correct answers.

1. \( \mu = np = 72 \), \( \sigma = \sqrt{np(1 - p)} = \sqrt{28.8} = 5.366 \).
2. Apply continuity correction: \( P(X \geq 80) \approx P(Y \geq 79.5) \).
3. Standardise: \( z = \frac{79.5 - 72}{5.366} = 1.398 \).
4. \( P(X \geq 80) \approx 1 - \Phi(1.398) \approx 0.081 \).

Inverse Normal Problems

• Typical prompt: find \( x \) such that \( P(X \leq x) = p \).
• Use \( x = \mu + z_p \sigma \), where \( z_p \) satisfies \( \Phi(z_p) = p \).

Example -- Percentile target

Chemistry practical scores obey \( X \sim \mathcal{N}(68.5, 4.2^2) \). What is the minimum score for the top 5%?

1. Need \( p = 0.95 \), so \( z_{0.95} = 1.645 \) (from tables or invNorm(0.95, 0, 1)).
2. \( x = 68.5 + 1.645 \times 4.2 \approx 75.4 \).
3. Scores \( \geq 75.4 \) lie in the top 5%.

Calculator Workflows

• Casio (fx-CG / fx-9860): NORMALCD(a, b, μ, σ) returns \( P(a \leq X \leq b) \); set a = -1E99 for \( -\infty \).
• TI (84/89): normalcdf(lower, upper, μ, σ) and invNorm(area, μ, σ) cover forward and inverse queries.
• Document the command in working: e.g. normalcdf(79.5, 10^99, 72, √28.8) for the continuity example.

Exam Watch Points

• Always draw the diagram first; SEAB mark schemes allocate method marks for the sketch.
• Quote the standardisation line explicitly, e.g. “Let \( Z = \frac{X - 72}{\sqrt{28.8}} \sim \mathcal{N}(0, 1) \).”
• When approximating, justify the conditions \( np \geq 5 \) and \( n(1 - p) \geq 5 \).
• Round intermediate \( z \)-values to at least 3 decimal places; round the final probability at the end.

Quick Revision Checklist

• [ ] Convert raw bounds to \( z \)-scores confidently and use symmetry shortcuts.
• [ ] Execute continuity corrections (\( \pm 0.5 \)) correctly for “at least/at most/exactly” statements.
• [ ] Use inverse normal commands to retrieve critical values for percentiles and control limits.
• [ ] Explain modelling decisions: why normal instead of binomial/Poisson, and which assumptions were made.

Next steps: Practise mixed binomial-normal approximation questions and combine them with sampling distributions in Topic 6.4.