When a normal approximation is suitable | CAIE 9709 Probability & Statistics 1 Notes

Probability & Statistics 1

Mathematics9709

Both expected counts must be greater than 5

B(30, 0.5): np = 15, nq = 15

balanced bell shape · approximation permitted

B(30, 0.05): np = 1.5, nq = 28.5

strongly skewed · use exact binomial

A large n does not guarantee a normal approximation. Both expected successes np and expected failures nq must be greater than 5.

X\sim B(n,p),\qquad q=1-p

np>5\quad\text{and}\quad nq>5

$np$ is the expected number of successes and $nq$ is the expected number of failures. If either is too small, the binomial shape is too strongly skewed for this syllabus approximation.

Common mistake

The condition is strictly greater than 5. A value equal to 5 does not pass the stated syllabus rule. Passing both conditions means “approximation permitted”, not “must use normal”.

Match the binomial mean and variance

Use a separate continuous variable $Y$ for the approximation:

Y\sim N(np,npq)

The mean is $np$ , the variance is $npq$ , and the standard deviation used for standardisation is $\sqrt{npq}$ .

Worked example

Write the matching normal model

For $X\sim B(150,0.1)$ ,

np=15,\qquad nq=135

npq=13.5

Answer

Y\sim N(15,13.5)

Key idea

The discrete variable

X

has not become continuous.

Y

is a continuous model used to estimate probabilities for

X

Reject the approximation when one condition fails

If the conditions fail, use the exact binomial PD/CD method. A large value of $n$ alone is not enough. Even when they pass, exact Binomial PD/CD remains the method when no approximation is requested.

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What the question gives or asks	Method to choose
A continuous variable with X ~ N(μ, σ²)	Normal CD; no continuity correction
A binomial variable and an ordinary probability	Exact Binomial PD/CD
Use a normal or suitable approximation to a binomial	Check np and nq, then use a normal model and continuity correction

Your turnIndependent check [6 marks]

Choose the required method in each case. (a) If $X\sim B(80,0.4)$ , state the method you would use to find $P(X\le30)$ . (b) If $U\sim B(120,0.3)$ , a question asks for a suitable normal approximation to $P(U\ge42)$ . Decide whether it is permitted and state the matching normal model; do not calculate the probability yet. (c) Decide whether $V\sim B(40,0.05)$ may be approximated by a normal distribution.

Show worked answer

In (a), the question asks for an ordinary binomial probability, so use exact Binomial CD. The fact that $np=32$ and $nq=48$ pass does not change that instruction.

In (b), $np=36$ and $nq=84$ , so the requested approximation is permitted. Its variance is $npq=25.2$ . In (c), $np=2$ , so reject the normal approximation and use an exact binomial method.

(a) Exact Binomial CD.
(b) The approximation is permitted, with $Y\sim N(36,25.2)$ .
(c) Reject the approximation and use exact Binomial PD/CD.