| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed hypothesis test (lower tail, H₁: p < p₀) |
| Difficulty | Moderate -0.3 This is a straightforward application of binomial hypothesis testing with clear structure: part (i) is routine probability calculation using binomial tables, and part (ii) is a standard one-tailed test where the method is prescribed and the critical value comparison is straightforward. The question requires no novel insight—just following the standard hypothesis testing procedure taught in S1. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05b Hypothesis test for binomial proportion |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X < 4) = P(X \leq 3)\) | M1 | |
| \(= 0.1074 + 0.2684 + 0.3020 + 0.2013 = 0.8791\) | A1 | Accept 0.879 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: p = 0.2\), \(H_1: p < 0.2\) | B1 | Both hypotheses correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X \leq 2) = 0.0692 + 0.2182 \times ...\) | M1 | Correct method |
| \(P(X \leq 2) = 0.2061\) | A1 | |
| \(0.2061 > 0.05\), so not significant | M1 | Correct comparison |
| No evidence at 5% level of a reduction in proportion of imperfect bowls | A1 A1 | Conclusion in context |
# Question 3:
**(i)**
$X \sim B(10, 0.2)$
$P(X < 4) = P(X \leq 3)$ | M1 |
$= 0.1074 + 0.2684 + 0.3020 + 0.2013 = 0.8791$ | A1 | Accept 0.879
**(ii)**
$H_0: p = 0.2$, $H_1: p < 0.2$ | B1 | Both hypotheses correct
$X \sim B(20, 0.2)$ under $H_0$
$P(X \leq 2) = 0.0692 + 0.2182 \times ...$ | M1 | Correct method
$P(X \leq 2) = 0.2061$ | A1 |
$0.2061 > 0.05$, so not significant | M1 | Correct comparison
No evidence at 5% level of a reduction in proportion of imperfect bowls | A1 A1 | Conclusion in context
---3 Over a long period of time, 20\% of all bowls made by a particular manufacturer are imperfect and cannot be sold.\\
(i) Find the probability that fewer than 4 bowls from a random sample of 10 made by the manufacturer are imperfect.
The manufacturer introduces a new process for producing bowls. To test whether there has been an improvement, each of a random sample of 20 bowls made by the new process is examined. From this sample, 2 bowls are found to be imperfect.\\
(ii) Show that this does not provide evidence, at the $5 \%$ level of significance, of a reduction in the proportion of imperfect bowls. You should show your hypotheses and calculations clearly.
\hfill \mbox{\textit{OCR MEI S1 2006 Q3 [8]}}