| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | State test assumptions or distributions |
| Difficulty | Standard +0.3 This is a standard S2 hypothesis testing question covering routine binomial test procedures. Part (a) asks for textbook assumptions, (b)-(d) involve straightforward critical region calculations with tables, and (e) is a normal approximation test. All parts follow predictable exam patterns with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04d Normal approximation to binomial2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The vacuum tubes shatter independently | B1 | For one correct reason which must mention tube(s) and shatter/shattering, or 2 correct reasons not in context |
| The probability of a vacuum tube shattering is constant | B1 | For 2 correct reasons which must mention tube(s) and shatter/shattering at least once |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(C \sim B(15, 0.35)\) plus \(\left[P(C \leq 9)\right] = 0.0142\) or \(\left[P(C\leq 10)\right] = 0.0124\) or \(\left[P(C \leq 9)\right] = 0.9876\) | M1 | For using correct distribution to find awrt 0.0142 or awrt 0.0124 or awrt 0.988; allow \(B(15,0.35)\) written and one of awrt 0.014 or awrt 0.012 or awrt 0.99 |
| Critical regions \(\left[0, C \leq 1\right]\) or \(\left[10 \leq C \leq 15\right]\) | M1 | For lower CR or \(C \leq 1\) oe; or upper CR \(C\geq 10\) oe; do not allow CR written as a probability statement |
| \(\left[0,\right] C \leq 1\) and \(10 \leq C \left[\leq 15\right]\) plus \(P(C \leq 9) = 0.0142\) and \(P(C\geq 10) = 0.0124\) | A1 | For both CR correct with relevant probabilities (3 sf, must be seen in part (b)); do not allow CR as probability statement |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.0266\) | B1ft | For awrt 0.0266 or 2.66% or ft the sum of probabilities in (b) for their 2 critical regions; if no probabilities seen, answer must be 0.0266 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| [4 is not in the CR therefore] there is no evidence to reject Rowan's belief | B1ft | For correct statement consistent with CR; must mention Rowan/his/her; must include words highlighted in bold |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(F \sim B(40, 0.35)\) | ||
| \(H_0: p = 0.35\) and \(H_1: p < 0.35\) | B1 | For both hypotheses correct in terms of \(p\) or \(\pi\) |
| \(P(F \leq 8) = 0.0303\) or CR \(F \leq 8\) | M1A1 | For using or writing \(P(F \leq 8)\) or awrt 0.0303 |
| Sufficient evidence to reject \(H_0\) or significant or 8 lies in the Critical region | M1 | For correct conclusion — need not be in context; ft their probability or CR |
| There is sufficient evidence to support that the proportion of type \(B\) vacuum tubes that shatter when exposed to alternating high and low temperatures is less than 35% | A1 | For correct conclusion in context with words highlighted in bold; independent of hypotheses; allow probability/number/amount/35% for proportion; allow "decreased" for "less than 35%" |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The vacuum **tubes shatter** independently | B1 | For one correct reason which must mention tube(s) and shatter/shattering, or 2 correct reasons not in context |
| The probability of a vacuum **tube shattering** is constant | B1 | For 2 correct reasons which must mention tube(s) and shatter/shattering at least once |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $C \sim B(15, 0.35)$ plus $\left[P(C \leq 9)\right] = 0.0142$ or $\left[P(C\leq 10)\right] = 0.0124$ or $\left[P(C \leq 9)\right] = 0.9876$ | M1 | For using correct distribution to find awrt 0.0142 or awrt 0.0124 or awrt 0.988; allow $B(15,0.35)$ written **and** one of awrt 0.014 or awrt 0.012 or awrt 0.99 |
| Critical regions $\left[0, C \leq 1\right]$ or $\left[10 \leq C \leq 15\right]$ | M1 | For lower CR or $C \leq 1$ oe; or upper CR $C\geq 10$ oe; do not allow CR written as a probability statement |
| $\left[0,\right] C \leq 1$ and $10 \leq C \left[\leq 15\right]$ plus $P(C \leq 9) = 0.0142$ and $P(C\geq 10) = 0.0124$ | A1 | For both CR correct with relevant probabilities (3 sf, must be seen in part (b)); do not allow CR as probability statement |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.0266$ | B1ft | For awrt 0.0266 or 2.66% or ft the sum of probabilities in (b) for their 2 critical regions; if no probabilities seen, answer must be 0.0266 |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| [4 is not in the CR therefore] there is no evidence to reject **Rowan's** belief | B1ft | For correct statement consistent with CR; must mention **Rowan/his/her**; must include words highlighted in bold |
### Part (e):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $F \sim B(40, 0.35)$ | | |
| $H_0: p = 0.35$ and $H_1: p < 0.35$ | B1 | For both hypotheses correct in terms of $p$ or $\pi$ |
| $P(F \leq 8) = 0.0303$ or CR $F \leq 8$ | M1A1 | For using or writing $P(F \leq 8)$ or awrt 0.0303 |
| Sufficient evidence to reject $H_0$ or significant or 8 lies in the Critical region | M1 | For correct conclusion — need not be in context; ft their probability or CR |
| There is sufficient evidence to support that the **proportion** of type $B$ vacuum **tubes** that shatter when exposed to alternating high and low temperatures is less than 35% | A1 | For correct conclusion in context with words highlighted in bold; independent of hypotheses; allow probability/number/amount/35% for proportion; allow "decreased" for "less than 35%" |
---\begin{enumerate}
\item Rowan believes that $35 \%$ of type $A$ vacuum tubes shatter when exposed to alternating high and low temperatures.
\end{enumerate}
Rowan takes a random sample of 15 of these type $A$ vacuum tubes and uses a two-tailed test, at the $5 \%$ level of significance, to test his belief.\\
(a) Give two assumptions, in context, that Rowan needs to make for a binomial distribution to be a suitable model for the number of these type $A$ vacuum tubes that shatter when exposed to alternating high and low temperatures.\\
(b) Using a binomial distribution, find the critical region for the test.
You should state the probability of rejection in each tail, which should be as close as possible to 0.025\\
(c) Find the actual level of significance of the test based on your critical region from part (b)
Rowan records that in the latest batch of 15 type $A$ vacuum tubes exposed to alternating high and low temperatures, 4 of them shattered.\\
(d) With reference to part (b), comment on Rowan's belief. Give a reason for your answer.
Rowan changes to type $B$ vacuum tubes. He takes a random sample of 40 type $B$ vacuum tubes and finds that 8 of them shatter when exposed to alternating high and low temperatures.\\
(e) Test, at the $5 \%$ level of significance, whether or not there is evidence that the proportion of type $B$ vacuum tubes that shatter when exposed to alternating high and low temperatures is lower than $35 \%$\\
You should state your hypotheses clearly.
\hfill \mbox{\textit{Edexcel S2 2024 Q3 [12]}}