| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Type I/II errors and power of test |
| Type | Calculate probability of Type I error |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question covering standard S2 material: setting up hypotheses for a proportion test, calculating Type I error probability from a binomial distribution, constructing a normal approximation confidence interval, and interpreting results. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05c Significance levels: one-tail and two-tail5.03a Continuous random variables: pdf and cdf5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: p = 0.1\); \(H_1: p < 0.1\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(B(40, 0.1)\) stated or implied by use of | B1 | e.g. by \(^{40}C_x\) or \(0.9^p \times 0.1^q\ (p+q=40)\) |
| \(0.9^{40} + 40 \times 0.9^{39} \times 0.1\) | M1 | Correct working (if seen). If working not seen, M1 may be implied by 0.0805 |
| \(= 0.0805\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(z = 1.645\) | B1 | seen |
| \(\frac{6}{80} \pm z\sqrt{\frac{\frac{6}{80}\times\frac{(80-6)}{80}}{80}}\) | M1 | Formula of correct form. Must be a 'z' |
| \(= 0.0266\) to \(0.123\) (3 sfs) | A1 | Allow 0.03 to 0.12 or better. Must be an interval |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 10% (or manufacturer's claim) is within CI, hence no reason to question claim | B1 | FT Allow '10% is within CI, accept claim'. Must include both parts. No contradictions. FT their CI. Note if CI is centred on 0.1 allow ft 0.075 is within CI, accept claim |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: p = 0.1$; $H_1: p < 0.1$ | B1 | |
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $B(40, 0.1)$ stated or implied by use of | B1 | e.g. by $^{40}C_x$ or $0.9^p \times 0.1^q\ (p+q=40)$ |
| $0.9^{40} + 40 \times 0.9^{39} \times 0.1$ | M1 | Correct working (if seen). If working not seen, M1 may be implied by 0.0805 |
| $= 0.0805$ | A1 | |
## Question 5(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $z = 1.645$ | B1 | seen |
| $\frac{6}{80} \pm z\sqrt{\frac{\frac{6}{80}\times\frac{(80-6)}{80}}{80}}$ | M1 | Formula of correct form. Must be a 'z' |
| $= 0.0266$ to $0.123$ (3 sfs) | A1 | Allow 0.03 to 0.12 or better. Must be an interval |
## Question 5(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| 10% (or manufacturer's claim) is within CI, hence no reason to question claim | B1 | FT Allow '10% is within CI, accept claim'. Must include both parts. No contradictions. FT their CI. Note if CI is centred on 0.1 allow ft 0.075 is within CI, accept claim |5 The manufacturer of a certain type of biscuit claims that $10 \%$ of packets include a free offer printed on the packet. Jyothi suspects that the true proportion is less than $10 \%$. He plans to test the claim by looking at 40 randomly selected packets and, if the number which include the offer is less than 2 , he will reject the manufacturer's claim.\\
(i) State suitable hypotheses for the test.\\
(ii) Find the probability of a Type I error.\\
On another occasion Jyothi looks at 80 randomly selected packets and finds that exactly 6 include the free offer.\\
(iii) Calculate an approximate $90 \%$ confidence interval for the proportion of packets that include the offer.\\
(iv) Use your confidence interval to comment on the manufacturer's claim.\\
$6 X$ is a random variable with probability density function given by
$$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$
where $a$ and $b$ are constants.\\
\hfill \mbox{\textit{CAIE S2 2019 Q5 [8]}}