Moderate -0.8 This is a straightforward application of the normal approximation to the binomial distribution with continuity correction. It requires only identifying n=200, p=0.08, calculating mean and variance, applying continuity correction (P(X≥15) = P(X>14.5)), and standardizing. This is a standard textbook exercise with no problem-solving insight required, making it easier than average.
2 On a production line making toys, the probability of any toy being faulty is 0.08 . A random sample of 200 toys is checked. Use a suitable approximation to find the probability that there are at least 15 faulty toys.
For standardising, with or without cc, must have \(\sqrt{}\) in denom
\(= \Phi(0.391)\)
M1
For use of continuity correction 14.5 or 15.5
\(= 0.652\)
M1 A1 [5]
For finding a prob \(> 0.5\) from their \(z\), legit; answer rounding to 0.652 c.w.o
## Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| mean $= 200 \times 0.08 = 16$, var $= 14.72$ | B1 | For both 16 and 14.7 seen |
| $P(X \geq 15) = 1 - \Phi\left(\frac{14.5-16}{\sqrt{14.72}}\right)$ | M1 | For standardising, with or without cc, must have $\sqrt{}$ in denom |
| $= \Phi(0.391)$ | M1 | For use of continuity correction 14.5 or 15.5 |
| $= 0.652$ | M1 A1 **[5]** | For finding a prob $> 0.5$ from their $z$, legit; answer rounding to 0.652 c.w.o |
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2 On a production line making toys, the probability of any toy being faulty is 0.08 . A random sample of 200 toys is checked. Use a suitable approximation to find the probability that there are at least 15 faulty toys.
\hfill \mbox{\textit{CAIE S1 2008 Q2 [5]}}