Moderate -0.3 This is a standard two-sample confidence interval calculation with summary statistics provided. Students must calculate sample means and variances from the summations, then apply the formula for a CI for difference of means. While it requires careful arithmetic and knowledge of the t-distribution, it's a routine procedure with no conceptual challenges or novel problem-solving required—slightly easier than average for Further Maths statistics.
1 Kayla is investigating the lengths of the leaves of a certain type of tree found in two forests \(X\) and \(Y\). She chooses a random sample of 40 leaves of this type from forest \(X\) and records their lengths, \(x \mathrm {~cm}\). She also records the lengths, \(y \mathrm {~cm}\), for a random sample of 60 leaves of this type from forest \(Y\). Her results are summarised as follows.
$$\sum x = 242.0 \quad \sum x ^ { 2 } = 1587.0 \quad \sum y = 373.2 \quad \sum y ^ { 2 } = 2532.6$$
Find a \(90 \%\) confidence interval for the difference between the population mean lengths of leaves in forests \(X\) and \(Y\).
1 Kayla is investigating the lengths of the leaves of a certain type of tree found in two forests $X$ and $Y$. She chooses a random sample of 40 leaves of this type from forest $X$ and records their lengths, $x \mathrm {~cm}$. She also records the lengths, $y \mathrm {~cm}$, for a random sample of 60 leaves of this type from forest $Y$. Her results are summarised as follows.
$$\sum x = 242.0 \quad \sum x ^ { 2 } = 1587.0 \quad \sum y = 373.2 \quad \sum y ^ { 2 } = 2532.6$$
Find a $90 \%$ confidence interval for the difference between the population mean lengths of leaves in forests $X$ and $Y$.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q1 [7]}}