| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2022 |
| Session | January |
| 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 two-part confidence interval question requiring standard formulas. Part (a) involves calculating a 95% CI from given data using known σ, while part (b) requires rearranging the width formula to find sample size for a 99% CI. Both are routine S3 procedures with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use of \(\bar{x} \pm z \times \frac{1.9}{\sqrt{10}}\) | M1 | For use of correct expression with 1.9, 10 and \(1 < z < 3\) |
| \(z = 1.96\) | B1 | For \(z = 1.96\) |
| \((52.54..., 54.897...)\) | A1 | For awrt 52.5 |
| awrt 52.5 and 54.9 | A1 | For awrt 54.9 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use of \(1.5 > 2 \times z \times \frac{1.9}{\sqrt{n}}\) | M1 | Use of \(z \times \frac{1.9}{\sqrt{n}}\) in a correct inequality with 0.75 or 1.5 and \(2 < z < 3\) (allow written as equation) |
| \(z = 2.5758\) (or better) | B1 | For \(z = 2.5758\) or better |
| \(1.5 > \frac{9.78804}{\sqrt{n}}\) | dM1 | Dependent on 1st M1, for solving a correct inequality for the width of the 99% CI (allow equation rather than inequality) |
| \(n > 42.58...\) so \(n = 43\) | A1 | cao |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of $\bar{x} \pm z \times \frac{1.9}{\sqrt{10}}$ | M1 | For use of correct expression with 1.9, 10 and $1 < z < 3$ |
| $z = 1.96$ | B1 | For $z = 1.96$ |
| $(52.54..., 54.897...)$ | A1 | For awrt 52.5 |
| awrt 52.5 and 54.9 | A1 | For awrt 54.9 |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of $1.5 > 2 \times z \times \frac{1.9}{\sqrt{n}}$ | M1 | Use of $z \times \frac{1.9}{\sqrt{n}}$ in a correct inequality with 0.75 or 1.5 and $2 < z < 3$ (allow written as equation) |
| $z = 2.5758$ (or better) | B1 | For $z = 2.5758$ or better |
| $1.5 > \frac{9.78804}{\sqrt{n}}$ | dM1 | Dependent on 1st M1, for solving a correct inequality for the width of the 99% CI (allow equation rather than inequality) |
| $n > 42.58...$ so $n = 43$ | A1 | cao |
---2. Krishi owns a farm on which he keeps chickens.
He selects, at random, 10 of the eggs produced and weighs each of them.\\
You may assume that these weights are a random sample from a normal distribution with standard deviation 1.9 g
The total weight of these 10 eggs is 537.2 g
\begin{enumerate}[label=(\alph*)]
\item Find a $95 \%$ confidence interval for the mean weight of the eggs produced by Krishi's chickens.
Krishi was hoping to obtain a $99 \%$ confidence interval of width at most 1.5 g
\item Calculate the minimum sample size necessary to achieve this.\\
\includegraphics[max width=\textwidth, alt={}, center]{fc43aabf-ad04-4852-8539-981cef608f31-04_2662_95_107_1962}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2022 Q2 [8]}}