Moderate -0.8 This is a straightforward application of the normal approximation to the binomial distribution with a single probability calculation. It requires identifying n=80, p=0.32, applying the continuity correction for P(X<20), standardizing to find a z-score, and reading from tables. The mechanics are routine with no conceptual challenges beyond standard S1 content.
2 In a large college, \(32 \%\) of the students have blue eyes. A random sample of 80 students is chosen. Use an approximation to find the probability that fewer than 20 of these students have blue eyes.
Substituting *their* 25.6 and 17.408 into \(\pm\)standardisation formula (any number for 19.5), not \(\sigma^2\), \(\sqrt{\sigma}\)
M1
Using continuity correction 19.5 or 20.5 in *their* standardisation formula
\(= [1 - \Phi(1.462)] = 1 - 0.9282\)
M1
Appropriate area \(\Phi\), from final process, must be probability. (Expect final ans \(< 0.5\)). Note: the correct final answer may imply M1 from use of calculator.
\(0.0718\)
A1
\(0.0718 \leqslant p \leqslant 0.0719\)
5
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Mean $= 80 \times 0.32 = 25.6$, $\text{var} = 80 \times 0.32 \times 0.68 = 17.408$ | **B1** | 25.6 and 17.4[08] seen, allow unsimplified. 4.172… implies correct variance. |
| $P(X < 20) = P\left(Z < \frac{19.5 - 25.6}{\sqrt{17.408}}\right) = P(Z < -1.462)$ | **M1** | Substituting *their* 25.6 and 17.408 into $\pm$standardisation formula (any number for 19.5), not $\sigma^2$, $\sqrt{\sigma}$ |
| | **M1** | Using continuity correction 19.5 or 20.5 in *their* standardisation formula |
| $= [1 - \Phi(1.462)] = 1 - 0.9282$ | **M1** | Appropriate area $\Phi$, from final process, must be probability. (Expect final ans $< 0.5$). Note: the correct final answer may imply M1 from use of calculator. |
| $0.0718$ | **A1** | $0.0718 \leqslant p \leqslant 0.0719$ |
| | **5** | |
2 In a large college, $32 \%$ of the students have blue eyes. A random sample of 80 students is chosen. Use an approximation to find the probability that fewer than 20 of these students have blue eyes.\\
\hfill \mbox{\textit{CAIE S1 2022 Q2 [5]}}