Moderate -0.8 Part (a) is pure recall of standard conditions (n large, p small, np moderate). Part (b) is a routine hypothesis test application with clearly signposted approximation and significance level. The setup is straightforward with no conceptual surprises, making this easier than average for A-level.
3. (a) Write down two conditions needed to approximate the binomial distribution by the Poisson distribution.
A machine which manufactures bolts is known to produce \(3 \%\) defective bolts. The machine breaks down and a new machine is installed. A random sample of 200 bolts is taken from those produced by the new machine and 12 bolts were defective.
(b) Using a suitable approximation, test at the \(5 \%\) level of significance whether or not the proportion of defective bolts is higher with the new machine than with the old machine. State your hypotheses clearly.
1st B1 for \(H_0\), 2nd B1 for \(H_1\); SC: if both correct but different letter used, award B1 B0; also allow B1 B0 for \(H_0: \lambda = 6\), \(H_1: \lambda > 6\)
M1 for writing or using \(1 - P(X \leq 11)\) or giving \(P(X \leq 10) = 0.9574\) or \(P(X \geq 11) = 0.0426\); 1st A1 for 0.0201 or CR \(X \geq 11/X > 10\); NB \(P(X \leq 11) = 0.9799\) on its own scores M1A1
\((0.0201 < 0.05)\) Reject \(H_0\) or Significant or 12 lies in the Critical region
M1 dep
2nd M1 dependent on 1st M1; correct statement based on table; do not allow non-contextual conflicting statements
There is evidence that the proportion of defective bolts has increased
A1 ft
2nd A1 ft for correct contextualised statement; NB a correct contextual statement on its own scores M1A1
# Question 3:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $n$ – large/high/big/ $n > 50$ | B1 | |
| $p$ – small/close to 0 / $p < 0.2$ | B1 | |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: p = 0.03$, $H_1: p > 0.03$ | B1, B1 | 1st B1 for $H_0$, 2nd B1 for $H_1$; SC: if both correct but different letter used, award B1 B0; also allow B1 B0 for $H_0: \lambda = 6$, $H_1: \lambda > 6$ |
| Po(6) | B1 | B1 writing or using Po(6) |
| $P(X \geq 12) = 1 - P(X \leq 11) = 1 - 0.9799 = 0.0201$ or $P(X \leq 10) = 0.9574$, $P(X \geq 11) = 0.0426$, CR $X \geq 11$ | M1, A1 | M1 for writing or using $1 - P(X \leq 11)$ or giving $P(X \leq 10) = 0.9574$ or $P(X \geq 11) = 0.0426$; 1st A1 for 0.0201 or CR $X \geq 11/X > 10$; NB $P(X \leq 11) = 0.9799$ on its own scores M1A1 |
| $(0.0201 < 0.05)$ Reject $H_0$ or Significant or 12 lies in the Critical region | M1 dep | 2nd M1 dependent on 1st M1; correct statement based on table; do not allow non-contextual conflicting statements |
| There is evidence that the proportion of defective bolts has increased | A1 ft | 2nd A1 ft for correct contextualised statement; NB a correct contextual statement on its own scores M1A1 |
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3. (a) Write down two conditions needed to approximate the binomial distribution by the Poisson distribution.
A machine which manufactures bolts is known to produce $3 \%$ defective bolts. The machine breaks down and a new machine is installed. A random sample of 200 bolts is taken from those produced by the new machine and 12 bolts were defective.\\
(b) Using a suitable approximation, test at the $5 \%$ level of significance whether or not the proportion of defective bolts is higher with the new machine than with the old machine. State your hypotheses clearly.\\
\hfill \mbox{\textit{Edexcel S2 2012 Q3 [9]}}