| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2021 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI from summary stats |
| Difficulty | Standard +0.3 This is a straightforward confidence interval question with known standard deviation requiring standard formula application. Part (a) is routine calculation, part (b) tests interpretation, and part (c) requires rearranging the confidence interval formula to find n—all standard S3 techniques with no novel problem-solving required. Slightly above average difficulty due to the algebraic manipulation in part (c). |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2.977 \pm 2.5758 \times \frac{0.015}{3}\) | M1, B1 | \(2.977 \pm (z\text{ value}) \times \frac{0.015}{3}\); awrt 2.5758 |
| \(= (2.9641\ldots,\ 2.9898\ldots)\) | A1 | awrt (2.964, 2.990) (condone 2.99) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The CI does not contain the stated weight | B1 | cao this must be consistent with their confidence interval |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2.995 - 1.96 \times \frac{0.015}{\sqrt{n}} < 2.991\) | M1 | Setting up an inequality using \(z\) value \(> 1.5\). Condone \(=\) |
| \(\sqrt{n} < \frac{1.96 \times 0.015}{2.995 - 2.991}\) | M1d | Dep on previous M mark. Correct rearranging to get \(\sqrt{n} < \ldots\) or \(n < \ldots\) Condone \(=\) or \(>\) |
| \(\sqrt{n} <\) awrt 7.35 | A1 | awrt 7.35 may be implied by awrt 54 |
| \(n = 54\) | A1cao | 54 |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2.977 \pm 2.5758 \times \frac{0.015}{3}$ | M1, B1 | $2.977 \pm (z\text{ value}) \times \frac{0.015}{3}$; awrt 2.5758 |
| $= (2.9641\ldots,\ 2.9898\ldots)$ | A1 | awrt (2.964, 2.990) (condone 2.99) |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The CI does not contain the stated weight | B1 | cao this must be consistent with their confidence interval |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2.995 - 1.96 \times \frac{0.015}{\sqrt{n}} < 2.991$ | M1 | Setting up an inequality using $z$ value $> 1.5$. Condone $=$ |
| $\sqrt{n} < \frac{1.96 \times 0.015}{2.995 - 2.991}$ | M1d | Dep on previous M mark. Correct rearranging to get $\sqrt{n} < \ldots$ or $n < \ldots$ Condone $=$ or $>$ |
| $\sqrt{n} <$ awrt 7.35 | A1 | awrt 7.35 may be implied by awrt 54 |
| $n = 54$ | A1cao | 54 |
---\begin{enumerate}
\item Assam produces bags of flour. The stated weight printed on the bags of flour is 3 kg . The weights of the bags of flour are normally distributed with standard deviation 0.015 kg .
\end{enumerate}
Assam weighs a random sample of 9 bags of flour and finds their mean weight is 2.977 kg .\\
(a) Calculate the $99 \%$ confidence interval for the mean weight of a bag of flour. Give your limits to 3 decimal places.
Assam decides to increase the amount of flour put into the bags.\\
(b) Explain why the confidence interval has led Assam to take this action.
After the increase a random sample of $n$ bags of flour is taken. The sample mean weight of these $n$ bags is 2.995 kg . A $95 \%$ confidence interval for $\mu$ gave a lower limit of less than 2.991 kg .\\
(c) Find the maximum value of $n$.\\
VILV SIHI NI IIII M I ON OC\\
VIAV SIHI NI III IM I ON OO\\
VIAV SIHI NI III HM ION OC
\hfill \mbox{\textit{Edexcel S3 2021 Q5 [8]}}