| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (lower tail, H₁: p < p₀) |
| Difficulty | Standard +0.3 This is a straightforward one-tailed binomial hypothesis test with standard steps (state hypotheses, find critical region, compare, conclude). The calculations are routine (n=12, p=0.4) and all parts follow textbook procedures. Part (iv) requires basic understanding of significance levels but no deep insight. Slightly easier than average due to small sample size making calculations simple and no unusual complications. |
| Spec | 2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Sample is random | B1 | Indicate random sample. Allow "unbiased sample" or "randomly selected" or "all equally likely". Allow "representative" provided it's clearly "of company" (not city). Not just "independent". Withhold if extra wrong bits. |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| List population, number sequentially | B1 | List can be implied; must imply employees or people. "Sequential" can be assumed. |
| Select using random numbers | B1 | Not "select numbers randomly". Don't need "ignore outside range" etc. Number randomly *and* select randomly, B1, but "assign random nos & arrange", B2. SC: Put names into hat/lottery machine and take them out: B2. SC: Systematic: B1 for list, can get second B1 if starting-point random. |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: p = 0.4\); \(H_1: p < 0.4\) | B2 | Both correct, B2. Allow \(\pi\). One error e.g. \(\mu\) or no symbol, B1, but \(\bar{x}\), \(z\) etc: B0. |
| \(B(12, 0.4)\) | M1 | \(B(12, 0.4)\) stated or implied. Can be implied by \(N(4.8, 2.88)\) but no further marks. 0.1673, 0.0398, 0.1513, 0.0421: M1A0(A1M1A1) |
| \(\alpha\): \(P(\leq 2) = \mathbf{0.0834}\) | A1 | \(P(\leq 2) = 0.0834\), or \(P(>2) = 0.9166\) |
| \(> 0.05\) | A1 | Compare numerical \(P(\leq 2)\) with 0.05, or \(P(>2)\) with 0.95 |
| \(\beta\): CR is \(\leq 1\) | A1 | CR is \(\leq 1\) stated. |
| 0.0196 seen and compare 2 with \(\leq 1\) | A1 | Explicitly compare 2 with CR; probability 0.0196 must be seen |
| Do not reject \(H_0\). | M1 | Correct first conclusion, needs \(P(\leq 2 \mid p = 0.4)\) or fully consistent equivalent |
| Insufficient evidence that proportion of employees from group Z is less. | A1ft | In context (mention "employees", "city" etc), acknowledge uncertainty ("evidence"). *Not* "there is evidence that the proportion of employees is 0.4". FT on wrong \(p\)-value or wrong critical value if previous mark gained. SC: Normal: B2 M1 max. SC: \(P(=2)\) or \(P(\geq 2)\) or \(P(<2)\): B2 M1 max. SC: two-tailed: can get B1B0 M1A1A0 M1A1 (don't give second A1 for 0.05) |
| [7] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Yes as \(H_0\) is rejected | M1 | Realise this changes conclusion (FT!), or "more likely to reject \(H_0\)", "larger CR" |
| A1 [2] | More supportive [just "more supportive" without evidence is M0A0] |
## Question 6:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Sample is random | B1 | Indicate random sample. Allow "unbiased sample" or "randomly selected" or "all equally likely". Allow "representative" provided it's clearly "of company" (not city). Not just "independent". Withhold if extra wrong bits. |
| | **[1]** | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| List population, number sequentially | B1 | List can be implied; must imply employees or people. "Sequential" can be assumed. |
| Select using random numbers | B1 | Not "select numbers randomly". Don't need "ignore outside range" etc. Number randomly *and* select randomly, B1, but "assign random nos & arrange", B2. SC: Put names into hat/lottery machine and take them out: B2. SC: Systematic: B1 for list, can get second B1 if starting-point random. |
| | **[2]** | |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: p = 0.4$; $H_1: p < 0.4$ | B2 | Both correct, B2. Allow $\pi$. One error e.g. $\mu$ or no symbol, B1, but $\bar{x}$, $z$ etc: B0. |
| $B(12, 0.4)$ | M1 | $B(12, 0.4)$ stated or implied. Can be implied by $N(4.8, 2.88)$ but no further marks. 0.1673, 0.0398, 0.1513, 0.0421: M1A0(A1M1A1) |
| $\alpha$: $P(\leq 2) = \mathbf{0.0834}$ | A1 | $P(\leq 2) = 0.0834$, or $P(>2) = 0.9166$ |
| $> 0.05$ | A1 | Compare numerical $P(\leq 2)$ with 0.05, or $P(>2)$ with 0.95 |
| $\beta$: CR is $\leq 1$ | A1 | CR is $\leq 1$ stated. |
| 0.0196 seen and compare 2 with $\leq 1$ | A1 | Explicitly compare 2 with CR; probability 0.0196 must be seen |
| Do not reject $H_0$. | M1 | Correct first conclusion, needs $P(\leq 2 \mid p = 0.4)$ or fully consistent equivalent |
| Insufficient evidence that proportion of employees from group Z is less. | A1ft | In context (mention "employees", "city" etc), acknowledge uncertainty ("evidence"). *Not* "there is evidence that the proportion of employees is 0.4". FT on wrong $p$-value or wrong critical value if previous mark gained. SC: Normal: B2 M1 max. SC: $P(=2)$ or $P(\geq 2)$ or $P(<2)$: B2 M1 max. SC: two-tailed: can get B1B0 M1A1A0 M1A1 (don't give second A1 for 0.05) |
| | **[7]** | |
### Part (iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Yes as $H_0$ is rejected | M1 | Realise this changes conclusion (FT!), or "more likely to reject $H_0$", "larger CR" |
| | A1 **[2]** | More supportive [just "more supportive" without evidence is M0A0] |
---6 In a city the proportion of inhabitants from ethnic group $\mathbf { Z }$ is known to be $\mathbf { 0 . 4 }$. A sample of $\mathbf { 1 2 }$ employees of a large company in this city is obtained and it is found that 2 of them are from ethnic group $Z$. A test is carried out, at the $5 \%$ significance level, of whether the proportion of employees in this company from ethnic group $Z$ is less than in the city as a whole.\\[0pt]
(i) State an assumption that must be made about the sample for a significance test to be valid. [1]\\[0pt]
(ii) Describe briefly an appropriate way of obtaining the sample. [2]\\[0pt]
(iii) Carry out the test. [7]\\
(iv) A manager believes that the company discriminates against ethnic group $Z$. Explain whether carrying out the test at the 10\% significance level would be more supportive or less supportive of the manager's belief. [2]
\hfill \mbox{\textit{OCR S2 2014 Q6 [12]}}