| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 19 |
| 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 standard S1 hypothesis testing question requiring binomial probability calculations and setting up a one-tailed test. Part (i) involves routine binomial probability calculations. Part (ii) is a textbook one-tailed test with clear hypotheses and straightforward comparison to significance level. Part (iii) extends to finding a two-tailed critical region, which requires more calculation but follows standard procedures. The question is slightly easier than average because it's well-structured with clear guidance, uses standard significance levels, and involves no conceptual surprises—just methodical application of learned techniques. |
| 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_1: p \neq 0.8\) | B1 | for \(H_1\) |
| Lower Tail: \(P(X \leq 10) = 0.0163 < 2.5\%\) | B1 | for 0.0163 or 0.0513 seen |
| \(P(X \leq 11) = 0.0513 > 2.5\%\) | M1dep | for either correct comparison with 2.5% (not 5%) seen or clearly implied |
| Correct lower tail CR (must have zero) | A1dep | for correct lower tail CR |
| Upper Tail: \(P(X \geq 17) = 1 - P(X \leq 16) = 1 - 0.9009 = 0.0991 > 2.5\%\) | B1 | for 0.0991 or 0.0180 seen |
| \(P(X \geq 18) = 1 - P(X \leq 17) = 1 - 0.9820 = 0.0180 < 2.5\%\) | M1dep | for either correct comparison with 2.5% (not 5%) seen or clearly implied |
| Critical region is \(\{0,1,2,3,4,5,6,7,8,9,10,18\}\) | A1dep | for correct upper tail CR |
| Correct CR without supportive working scores SC2 max after 1st B1 (SC1 for each fully correct tail of CR) | Total: 7 |
## Question 1 (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_1: p \neq 0.8$ | B1 | for $H_1$ |
| **Lower Tail:** $P(X \leq 10) = 0.0163 < 2.5\%$ | B1 | for 0.0163 or 0.0513 seen |
| $P(X \leq 11) = 0.0513 > 2.5\%$ | M1dep | for either correct comparison with **2.5%** (not 5%) seen or clearly implied |
| Correct lower tail CR (must have zero) | A1dep | for correct lower tail CR |
| **Upper Tail:** $P(X \geq 17) = 1 - P(X \leq 16) = 1 - 0.9009 = 0.0991 > 2.5\%$ | B1 | for 0.0991 or 0.0180 seen |
| $P(X \geq 18) = 1 - P(X \leq 17) = 1 - 0.9820 = 0.0180 < 2.5\%$ | M1dep | for either correct comparison with **2.5%** (not 5%) seen or clearly implied |
| Critical region is $\{0,1,2,3,4,5,6,7,8,9,10,18\}$ | A1dep | for correct upper tail CR |
| Correct CR without supportive working scores SC2 max after 1st B1 (SC1 for each fully correct tail of CR) | | **Total: 7** |
---1 An online shopping company takes orders through its website. On average $80 \%$ of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that\\
(A) exactly 8 of these orders are delivered within 24 hours,\\
(B) at least 8 of these orders are delivered within 24 hours.
The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than $80 \%$ of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
\item Write down suitable hypotheses and carry out a test at the $5 \%$ significance level to determine whether there is any evidence to support the quality controller's suspicion.
\item A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the $5 \%$ significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18, find the critical region for this test, showing all of your calculations.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S1 Q1 [19]}}