Porisma

Worked solution

Approximate a binomial sample mean and track its endpoints

Fifty random observations are taken from B(20,0.2)B(20,0.2). Find the probability that the sample mean lies between 3.8 and 4.2. marks

[2009/IJC/II/12(a)]

CJC · Review problems · Question 3

Answer

Printed CLT approximation 0.5710.571; strict-endpoint exact value 0.5473642330.547364233 (inclusive endpoints: 0.5935239520.593523952).

Printed CLT calculation

Each observation has mean 4 and variance 20(0.2)(0.8)=3.220(0.2)(0.8)=3.2. For 50 independent observations, Xˉ∼˙N(4,3.2/50)=N(4,0.064)\bar X\mathrel{\dot\sim}N(4,3.2/50)=N(4,0.064) by the CLT. The printed working uses P(3.8<Xˉ<4.2)≈0.571P(3.8<\bar X<4.2)\approx0.571, to 3 s.f.

How did this part go?

Authored exact endpoint check

The total is exactly T∼B(1000,0.2)T\sim B(1000,0.2). With the strict inequalities used in the printed working, 190<T<210190<T<210, hence 191≤T≤209191\le T\le209 and the exact probability is about 0.547364233. If “between” is read inclusively, the exact event is 190≤T≤210190\le T\le210 with probability about 0.593523952. Endpoints have zero probability only in the continuous normal approximation; the source’s 0.571 remains labelled as an unadjusted CLT result.

How did this part go?

Common mistakes

The discrete sample mean has positive probability at its endpoints. State the endpoint convention when giving an exact binomial comparison.

Solution notes

“Between” is not explicitly inclusive in the prompt; the printed working uses strict inequalities. The authored exact comparisons retain that ambiguity rather than silently choosing an exact endpoint convention.

Read the hint