1 A factory produces small bottles of natural spring water. Two different machines, \(X\) and \(Y\), are used to fill empty bottles with the water. A quality control engineer checks the volumes of water in the bottles filled by each of the machines. He chooses a random sample of 60 bottles filled by machine \(X\) and a random sample of 75 bottles filled by machine \(Y\). The volumes of water, \(x\) and \(y\) respectively, in millilitres, are summarised as follows. $$\sum x = 6345 \quad \sum ( x - \bar { x } ) ^ { 2 } = 243.8 \quad \sum y = 7614 \quad \sum ( y - \bar { y } ) ^ { 2 } = 384.9$$ \(\bar { x }\) and \(\bar { y }\) are the sample means of the volume of water in the bottles filled by machines \(X\) and \(Y\) respectively. Find a \(95 \%\) confidence interval for the difference between the mean volume of water in bottles filled by machine \(X\) and the mean volume of water in bottles filled by machine \(Y\).

## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $s_x^2 = \frac{243.8}{59} \left[= \frac{1219}{295} = 4.132\right]$, $s_y^2 = \frac{384.9}{74} \left[= \frac{3849}{740} = 5.201\right]$ | **B1** | Implied by correct $s$ or pooled estimate $\frac{243.8 + 384.9}{60 + 75 - 2} = 4.727$ |
| $s^2 = \frac{4.132}{60} + \frac{5.201}{75} \left[= 0.1382\right]$ **or** $s = 0.3718$ | **M1** | Using their sample variances. Pooled estimate M0. |
| | **A1** | |
| CI: $\frac{6345}{60} - \frac{7614}{75} \pm 1.96 \times \text{'}0.3718\text{'}$ | **M1** | With a $z$-value |
| | **A1** | With 1.96 (with their $s$). Pooled: $\frac{6345}{60} - \frac{7614}{75} \pm 1.96 \times 2.174\sqrt{\frac{1}{60} + \frac{1}{75}}$ |
| $\left[3.5[0], 4.96\right]$ or $(3.5, 4.96)$ | **A1** | $4.23 \pm 0.729$ is A0, condone $[4.96, 3.5]$ etc. |
| | **Total: 6** | |

1 A factory produces small bottles of natural spring water. Two different machines, $X$ and $Y$, are used to fill empty bottles with the water. A quality control engineer checks the volumes of water in the bottles filled by each of the machines. He chooses a random sample of 60 bottles filled by machine $X$ and a random sample of 75 bottles filled by machine $Y$. The volumes of water, $x$ and $y$ respectively, in millilitres, are summarised as follows.

$$\sum x = 6345 \quad \sum ( x - \bar { x } ) ^ { 2 } = 243.8 \quad \sum y = 7614 \quad \sum ( y - \bar { y } ) ^ { 2 } = 384.9$$

$\bar { x }$ and $\bar { y }$ are the sample means of the volume of water in the bottles filled by machines $X$ and $Y$ respectively.

Find a $95 \%$ confidence interval for the difference between the mean volume of water in bottles filled by machine $X$ and the mean volume of water in bottles filled by machine $Y$.\\

\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q1 [6]}}