4 The number, \(x\), of a certain type of sea shell was counted at 60 randomly chosen sites, each one metre square, along the coastline in country \(A\). The number, \(y\), of the same type of sea shell was counted at 50 randomly chosen sites, each one metre square, along the coastline in country \(B\). The results are summarised as follows, where \(\bar{x}\) and \(\bar{y}\) denote the sample means of \(x\) and \(y\) respectively. $$\bar{x} = 29.2 \quad \Sigma(x - \bar{x})^{2} = 4341.6 \quad \bar{y} = 24.4 \quad \Sigma(y - \bar{y})^{2} = 3732.0$$ Find a \(95\%\) confidence interval for the difference between the mean number of sea shells, per square metre, on the coastlines in country \(A\) and in country \(B\).

## Question 4:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Sample variances are $\frac{4341.6}{59}$ and $\frac{3732.0}{49}$ | 1 | M1 | Either attempted |
| $s_x^2 = 73.59$ and $s_y^2 = 76.16$ | 1 | A1 | Both correct to at least 2 dp |
| Variance for difference of means: $s^2 = \dfrac{s_x^2}{60} + \dfrac{s_y^2}{50}$ | 1 | M1 | |
| $s^2 = 2.750$ or $s = 1.658$ | 1 | A1 | Correct value to at least 4 sf |
| Evaluate $(\bar{x} - \bar{y}) \pm zs$ | 1 | M1 | Any $z$-value, and using their value of $s$ |
| Use of correct $z = 1.96$ | 1 | A1 | |
| End-points of confidence interval are 1.55 and 8.05 | 1 | A1 | Accept e.g. $(1.55, 8.05)$ or $1.55 < \mu_A - \mu_B < 8.05$ |
| **Total** | **7** | | |

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4 The number, $x$, of a certain type of sea shell was counted at 60 randomly chosen sites, each one metre square, along the coastline in country $A$. The number, $y$, of the same type of sea shell was counted at 50 randomly chosen sites, each one metre square, along the coastline in country $B$. The results are summarised as follows, where $\bar{x}$ and $\bar{y}$ denote the sample means of $x$ and $y$ respectively.

$$\bar{x} = 29.2 \quad \Sigma(x - \bar{x})^{2} = 4341.6 \quad \bar{y} = 24.4 \quad \Sigma(y - \bar{y})^{2} = 3732.0$$

Find a $95\%$ confidence interval for the difference between the mean number of sea shells, per square metre, on the coastlines in country $A$ and in country $B$.

\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q4 [7]}}