| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | Expected value and most likely value |
| Difficulty | Standard +0.3 This is a straightforward binomial distribution question covering standard S1 content: calculating probabilities, expectation, mode, and conducting a one-tailed hypothesis test. All parts follow textbook procedures with no novel problem-solving required. The critical region calculation is routine, though part (vi) requires understanding of significance levels. Slightly above average difficulty due to the multi-part nature and the conceptual element in (vi), but still a standard exam question. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05a Hypothesis testing language: null, alternative, p-value, significance2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \geq 4) = 1 - P(X \leq 3) = 1 - 0.5489 = 0.4511\) | B1 for \(0.5489\), M1 for \(1 -\) their \(0.5489\), A1 CAO | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = np = 17\times0.2 = 3.4\) | M1 for product, A1 CAO | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| So 3 applicants is most likely | B1 for \(0.2393\), B1 for \(0.2093\), A1 CAO dep on both B1s | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \((B)\) \(H_1\) has this form as the suggestion is that mathematics graduates are more likely to be successful. | B1 for definition of \(p\) in context, B1 for \(H_0\), B1 for \(H_1\), E1 | 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| So critical region is \(\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\) | B1 for \(0.1057\), B1 for \(0.0377\), M1 for at least one comparison with \(5\%\), A1 CAO for critical region dep on M1 and at least one B1 | 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Or comparison \(P(X \geq 7) = 0.0377 < 10\%\) | E1, E1 | 2 marks |
# Question 8:
## Part (i)
$X \sim B(17, 0.2)$
$P(X \geq 4) = 1 - P(X \leq 3) = 1 - 0.5489 = 0.4511$ | B1 for $0.5489$, M1 for $1 -$ their $0.5489$, A1 CAO | 3 marks
## Part (ii)
$E(X) = np = 17\times0.2 = 3.4$ | M1 for product, A1 CAO | 2 marks
## Part (iii)
$P(X=2) = 0.3096 - 0.1182 = 0.1914$
$P(X=3) = 0.5489 - 0.3096 = 0.2393$
$P(X=4) = 0.7582 - 0.5489 = 0.2093$
So 3 applicants is most likely | B1 for $0.2393$, B1 for $0.2093$, A1 CAO dep on both B1s | 3 marks
## Part (iv)
$(A)$ Let $p =$ probability of a randomly selected maths graduate applicant being successful (for population)
$H_0: p = 0.2$
$H_1: p > 0.2$
$(B)$ $H_1$ has this form as the suggestion is that mathematics graduates are more likely to be successful. | B1 for definition of $p$ in context, B1 for $H_0$, B1 for $H_1$, E1 | 4 marks
## Part (v)
Let $X \sim B(17, 0.2)$
$P(X \geq 6) = 1 - P(X \leq 5) = 1 - 0.8943 = 0.1057 > 5\%$
$P(X \geq 7) = 1 - P(X \leq 6) = 1 - 0.9623 = 0.0377 < 5\%$
So critical region is $\{7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}$ | B1 for $0.1057$, B1 for $0.0377$, M1 for at least one comparison with $5\%$, A1 CAO for critical region dep on M1 and at least one B1 | 4 marks
## Part (vi)
Because $P(X \geq 6) = 0.1057 > 10\%$
Either: comment that 6 is still outside the critical region
Or comparison $P(X \geq 7) = 0.0377 < 10\%$ | E1, E1 | 2 marks8 A multinational accountancy firm receives a large number of job applications from graduates each year. On average $20 \%$ of applicants are successful.
A researcher in the human resources department of the firm selects a random sample of 17 graduate applicants.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that at least 4 of the 17 applicants are successful.
\item Find the expected number of successful applicants in the sample.
\item Find the most likely number of successful applicants in the sample, justifying your answer.
It is suggested that mathematics graduates are more likely to be successful than those from other fields. In order to test this suggestion, the researcher decides to select a new random sample of 17 mathematics graduate applicants. The researcher then carries out a hypothesis test at the $5 \%$ significance level.
\item (A) Write down suitable null and alternative hypotheses for the test.\\
(B) Give a reason for your choice of the alternative hypothesis.
\item Find the critical region for the test at the $5 \%$ level, showing all of your calculations.
\item Explain why the critical region found in part (v) would be unaltered if a $10 \%$ significance level were used.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S1 2007 Q8 [18]}}