| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Calculate Type I error probability |
| Difficulty | Moderate -0.5 This is a straightforward application of hypothesis testing with a binomial distribution. Students need to identify H₀: p=1/8 (guessing) and H₁: p>1/8, then calculate P(X≥3) under H₀ using binomial probability. The concepts are standard S2 material with direct calculation required, though understanding Type I error and significance level adds slight conceptual depth beyond pure computation. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: P(\text{correct}) = \frac{1}{8}\), \(H_1: P(\text{correct}) > \frac{1}{8}\) | B1 [1] | Or \(H_0\ p = 1/8\), \(H_1\ p > 1/8\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(1-\left(\left(\frac{1}{8}\right)^{10}+10\left(\frac{1}{8}\right)^9\left(\frac{7}{8}\right)+{}^{10}C_2\left(\frac{1}{8}\right)^8\left(\frac{7}{8}\right)^2\right)\) | M1 | M1 for attempt at correct expression; accept 1 error only, e.g. 1 term extra, omitted or wrong, or omit "1−" or incorrect p/q |
| A1 | Correct expression | |
| \(= 0.120\) (3 sf) or \(0.119\) | A1 [3] | Note: Use of Poisson in (ii) could score M1 only for expression \(1-P(0,1,2)\ \lambda=1.25\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 12% | B1f [1] | ft their (ii); must be a probability |
## Question 2:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: P(\text{correct}) = \frac{1}{8}$, $H_1: P(\text{correct}) > \frac{1}{8}$ | B1 [1] | Or $H_0\ p = 1/8$, $H_1\ p > 1/8$ |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $1-\left(\left(\frac{1}{8}\right)^{10}+10\left(\frac{1}{8}\right)^9\left(\frac{7}{8}\right)+{}^{10}C_2\left(\frac{1}{8}\right)^8\left(\frac{7}{8}\right)^2\right)$ | M1 | M1 for attempt at correct expression; accept 1 error only, e.g. 1 term extra, omitted or wrong, or omit "1−" or incorrect p/q |
| | A1 | Correct expression |
| $= 0.120$ (3 sf) or $0.119$ | A1 [3] | Note: Use of Poisson in (ii) could score M1 only for expression $1-P(0,1,2)\ \lambda=1.25$ |
### Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 12% | B1f [1] | ft their (ii); must be a probability |
---2 Marie claims that she can predict the winning horse at the local races. There are 8 horses in each race. Nadine thinks that Marie is just guessing, so she proposes a test. She asks Marie to predict the winners of the next 10 races and, if she is correct in 3 or more races, Nadine will accept Marie's claim.\\
(i) State suitable null and alternative hypotheses.\\
(ii) Calculate the probability of a Type I error.\\
(iii) State the significance level of the test.
\hfill \mbox{\textit{CAIE S2 2015 Q2 [5]}}