4 A company has two different machines, \(X\) and \(Y\), each of which fills empty cups with coffee. The manager is investigating the volumes of coffee, \(x\) and \(y\), measured in appropriate units, in the cups filled by machines \(X\) and \(Y\) respectively. She chooses a random sample of 50 cups filled by machine \(X\) and a random sample of 40 cups filled by machine \(Y\). The volumes are summarised as follows. $$\sum x = 15.2 \quad \sum x ^ { 2 } = 5.1 \quad \sum y = 13.4 \quad \sum y ^ { 2 } = 4.8$$ The manager claims that there is no difference between the mean volume of coffee in cups filled by machine \(X\) and the mean volume of coffee in cups filled by machine \(Y\). Test the manager's claim at the \(10 \%\) significance level.

## Question 4:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu_x = \mu_y$, $H_1: \mu_x \neq \mu_y$ | B1 | |
| $s_x^2 = \frac{1}{49}\!\left(5.1 - \frac{15.2^2}{50}\right) = 0.0097796$; $s_y^2 = \frac{1}{39}\!\left(4.8 - \frac{13.4^2}{40}\right) = 0.007974$ | M1A1 | |
| $s^2 = \frac{0.00977959}{50} + \frac{0.007974}{40} = 0.0003949$ | M1A1 | |
| $z = \dfrac{0.304 - 0.335}{\sqrt{0.0003949}} = (-)1.56$ | M1A1 | |
| Compare with 1.645 | M1 | |
| Accept $H_0$: insufficient evidence to reject manager's claim | A1 | |

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4 A company has two different machines, $X$ and $Y$, each of which fills empty cups with coffee. The manager is investigating the volumes of coffee, $x$ and $y$, measured in appropriate units, in the cups filled by machines $X$ and $Y$ respectively. She chooses a random sample of 50 cups filled by machine $X$ and a random sample of 40 cups filled by machine $Y$. The volumes are summarised as follows.

$$\sum x = 15.2 \quad \sum x ^ { 2 } = 5.1 \quad \sum y = 13.4 \quad \sum y ^ { 2 } = 4.8$$

The manager claims that there is no difference between the mean volume of coffee in cups filled by machine $X$ and the mean volume of coffee in cups filled by machine $Y$.

Test the manager's claim at the $10 \%$ significance level.\\

\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q4 [9]}}