Standard +0.8 This is a Further Maths statistics question requiring construction of a confidence interval for the difference of two means using summary statistics. While the procedure is standard (pooled variance, t-distribution), it involves multiple steps including calculating standard error, finding critical values, and correct interpretation. The Further Maths context and need for careful handling of two-sample inference places it moderately above average difficulty.
2 A random sample of 40 observations of a random variable \(X\) and a random sample of 50 observations of a random variable \(Y\) are taken. The resulting values for the sample means, \(\bar { x }\) and \(\bar { y }\), and the unbiased estimates, \(\mathrm { s } _ { \mathrm { x } } ^ { 2 }\) and \(\mathrm { s } _ { \mathrm { y } } ^ { 2 }\), for the population variances are as follows.
$$\bar { x } = 24.4 \quad \bar { y } = 17.2 \quad s _ { x } ^ { 2 } = 10.2 \quad s _ { y } ^ { 2 } = 11.1$$
Find a \(90 \%\) confidence interval for the difference between the population means of \(X\) and \(Y\).
2 A random sample of 40 observations of a random variable $X$ and a random sample of 50 observations of a random variable $Y$ are taken. The resulting values for the sample means, $\bar { x }$ and $\bar { y }$, and the unbiased estimates, $\mathrm { s } _ { \mathrm { x } } ^ { 2 }$ and $\mathrm { s } _ { \mathrm { y } } ^ { 2 }$, for the population variances are as follows.
$$\bar { x } = 24.4 \quad \bar { y } = 17.2 \quad s _ { x } ^ { 2 } = 10.2 \quad s _ { y } ^ { 2 } = 11.1$$
Find a $90 \%$ confidence interval for the difference between the population means of $X$ and $Y$.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q2 [5]}}