| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2020 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | One-tailed test (increase or decrease) |
| Difficulty | Standard +0.8 This is a Further Maths Statistics question requiring multiple Poisson hypothesis test techniques including Type II error calculation, which goes beyond standard A-level. While the individual calculations are methodical, the multi-part structure, parameter scaling across different time periods, and Type II error probability (requiring understanding of both null and alternative distributions) make this moderately challenging for Further Maths students. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X \sim Po(24)\) | B1 | For realising the distribution is \(Po(24)\) (May be seen or implied in part (ii)) |
| \(P(X = 26) = 0.071912\ldots\) awrt \(\mathbf{0.0719}\) | B1 | awrt 0.0719 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(X \geq 21) = 1 - P(X \leq 20)\ [= 1 - 0.24263\ldots]\) | M1 | Writing or using \(1 - P(X \leq 20)\) |
| \(= 0.75736\ldots\) awrt \(\mathbf{0.757}\) | A1 | awrt 0.757 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \lambda = 2\ [\mu = 16]\), \(H_1: \lambda < 2\ [\mu < 16]\) | B1 | Both hypotheses correct (must use \(\mu\) or \(\lambda\)) |
| \(P(Y \leq 10 \mid Y \sim Po(16)) = 0.077396\ldots\) awrt \(\mathbf{0.0774}\) | B1 | awrt 0.0774; allow awrt 0.08 from correct probability statement; allow CR: \(X \leq 9\) |
| Not significant / Do not reject \(H_0\) / 10 is not in the CR | M1 | Correct non-contextual conclusion (may be implied by correct contextual conclusion). Allow f.t. comparison of 'their \(p\)' with 0.05. Ignore contradictory contextual comments for this mark |
| There is not sufficient evidence to suggest a decrease/change in the rate of customers entering Jeff's supermarket | A1 | Fully correct solution drawing correct inference in context with all previous marks in (b) scored |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use of \(Po(8)\) to attempt critical region | M1 | \([P(Y \leq 3) = 0.0423\ldots\ P(Y \leq 4) = 0.0996\ldots]\) |
| Critical region is \(Y \leq 3\); \(H_0\) is not rejected when \(Y \geq 4\) | A1 | Finding critical region for the test \(Y \leq 3\) which must come from \(Po(8)\) |
| True distribution is \(W \sim Po(4)\) | B1 | Identifying the need to use \(Po(4)\) as the true distribution. Allow \(Po(4)\) seen or used for this mark |
| \(P(W \geq 4 \mid W \sim Po(4)) = 1 - P(W \leq 3)\ [= 1 - 0.43347\ldots]\) | M1 | Writing or using \(P(W \geq \text{`}4\text{'})\) or \(1 - P(W \leq \text{`}3\text{'})\) from \(Po(4)\). Allow f.t. on their identified CR but must be using \(Po(4)\) |
| \(= 0.56652\ldots\) awrt \(\mathbf{0.567}\) | A1 | awrt 0.567 |
## Question 1:
### Part (a)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim Po(24)$ | B1 | For realising the distribution is $Po(24)$ (May be seen or implied in part (ii)) |
| $P(X = 26) = 0.071912\ldots$ awrt $\mathbf{0.0719}$ | B1 | awrt 0.0719 |
### Part (a)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(X \geq 21) = 1 - P(X \leq 20)\ [= 1 - 0.24263\ldots]$ | M1 | Writing or using $1 - P(X \leq 20)$ |
| $= 0.75736\ldots$ awrt $\mathbf{0.757}$ | A1 | awrt 0.757 |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \lambda = 2\ [\mu = 16]$, $H_1: \lambda < 2\ [\mu < 16]$ | B1 | Both hypotheses correct (must use $\mu$ or $\lambda$) |
| $P(Y \leq 10 \mid Y \sim Po(16)) = 0.077396\ldots$ awrt $\mathbf{0.0774}$ | B1 | awrt 0.0774; allow awrt 0.08 from correct probability statement; allow CR: $X \leq 9$ |
| Not significant / Do not reject $H_0$ / 10 is not in the CR | M1 | Correct non-contextual conclusion (may be implied by correct contextual conclusion). Allow f.t. comparison of 'their $p$' with 0.05. Ignore contradictory contextual comments for this mark |
| There is not sufficient evidence to suggest a decrease/change in the rate of customers entering Jeff's supermarket | A1 | Fully correct solution drawing correct inference in context **with all previous marks in (b) scored** |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of $Po(8)$ to attempt critical region | M1 | $[P(Y \leq 3) = 0.0423\ldots\ P(Y \leq 4) = 0.0996\ldots]$ |
| Critical region is $Y \leq 3$; $H_0$ is not rejected when $Y \geq 4$ | A1 | Finding critical region for the test $Y \leq 3$ which must come from $Po(8)$ |
| True distribution is $W \sim Po(4)$ | B1 | Identifying the need to use $Po(4)$ as the true distribution. Allow $Po(4)$ seen or used for this mark |
| $P(W \geq 4 \mid W \sim Po(4)) = 1 - P(W \leq 3)\ [= 1 - 0.43347\ldots]$ | M1 | Writing or using $P(W \geq \text{`}4\text{'})$ or $1 - P(W \leq \text{`}3\text{'})$ from $Po(4)$. Allow f.t. on their **identified** CR but must be using $Po(4)$ |
| $= 0.56652\ldots$ awrt $\mathbf{0.567}$ | A1 | awrt 0.567 |
---\begin{enumerate}
\item The number of customers entering Jeff's supermarket each morning follows a Poisson distribution.
\end{enumerate}
Past information shows that customers enter at an average rate of 2 every 5 minutes.\\
Using this information,\\
(a) (i) find the probability that exactly 26 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning,\\
(ii) find the probability that at least 21 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning.
A rival supermarket is opened nearby. Following its opening, the number of customers entering Jeff's supermarket over a randomly selected 40-minute period is found to be 10\\
(b) Test, at the 5\% significance level, whether or not there is evidence of a decrease in the rate of customers entering Jeff's supermarket. State your hypotheses clearly.
A further randomly selected 20 -minute period is observed and the hypothesis test is repeated. Given that the true rate of customers entering Jeff's supermarket is now 1 every 5 minutes,\\
(c) calculate the probability of a Type II error.
\hfill \mbox{\textit{Edexcel FS1 2020 Q1 [13]}}