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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 218

For student $i$ in school $j$, "lme1" is the <random-intercept linear mixed model>
$$
\operatorname{THK}_{ij}
=\beta_0+\beta_1\operatorname{PTHK}_{ij}
+\beta_2\operatorname{TV}_j+\beta_3\operatorname{SC}_j
+b_j+\varepsilon_{ij},
$$
where
$$
b_j\overset{\rm iid}\sim N(0,\tau^2),
\qquad
\varepsilon_{ij}\overset{\rm iid}\sim N(0,\sigma^2),
$$
independently. The estimates are
$$
(\widehat\beta_0,\widehat\beta_1,\widehat\beta_2,\widehat\beta_3)
=(1.78880,0.30973,0.02175,0.47023),
$$
$$
\widehat\tau^2=0.0437,
\qquad
\widehat\sigma^2=1.6531.
$$
The fixed intercept is the population-average expected post-study score for an untreated reference student with PTHK zero. The random intercept $b_j$ is school $j$'s deviation from that population intercept.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 218 / 4 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 218 / 4 / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 218 / 4 / a / Solution

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