Challenging +1.2 This is a straightforward application of confidence intervals for predicted mean response in regression. Students must find the regression line from summary statistics, calculate the predicted mean at x=20, then apply the standard formula for confidence interval of mean prediction using given variance. While it requires multiple steps (regression coefficients, standard error calculation, critical value), each step follows a standard procedure taught in Further Statistics with no novel insight required. The given variance simplifies what would otherwise be a more complex calculation.
9 The values of a set of bivariate data \(\left( x _ { i } , y _ { i } \right)\) can be summarised by
$$n = 50 , \sum x = 1270 , \sum y = 5173 , \sum x ^ { 2 } = 42767 , \sum y ^ { 2 } = 701301 , \sum x y = 173161 .$$
Ten independent observations of \(Y\) are obtained, all corresponding to \(x = 20\). It may be assumed that the variance of \(Y\) is 1.9 , independently of the value of \(x\). Find a \(95 \%\) confidence interval for the mean \(\bar { Y }\) of the 10 observations of \(Y\).
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9 The values of a set of bivariate data $\left( x _ { i } , y _ { i } \right)$ can be summarised by
$$n = 50 , \sum x = 1270 , \sum y = 5173 , \sum x ^ { 2 } = 42767 , \sum y ^ { 2 } = 701301 , \sum x y = 173161 .$$
Ten independent observations of $Y$ are obtained, all corresponding to $x = 20$. It may be assumed that the variance of $Y$ is 1.9 , independently of the value of $x$. Find a $95 \%$ confidence interval for the mean $\bar { Y }$ of the 10 observations of $Y$.
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\hfill \mbox{\textit{OCR Further Statistics 2018 Q9 [8]}}