= Solution
The S-Plus input and the data table differ in one entry: the former uses $33$ for the first year in the second male age group, whereas the table has $22$. The displayed coefficients are reproduced by the software vector with $33$, so that is the input analysed here. Using $22$ instead would give a different fit, including an intercept of $36.175$ rather than $37.275$.
The factor grid has $5\times4\times2=40$ rows. Its first factor varies fastest, so the five years run within each age group, and the age groups run within each sex. All three predictors are <regression factors>: in particular the year coefficients are category contrasts, not a fitted linear slope in calendar time. Under reference-level coding, take the first year, youngest age group and men as the <reference levels in a regression factor>. Write $s=1$ for women and $0$ for men, and $I_{yk},I_{aj}$ for the other year and age indicators. The <normal linear model> is
$$
P=\alpha+\sum_{k=2}^5\gamma_kI_{yk}+\lambda s+\sum_{j=2}^4\eta_jI_{aj}+\sum_{j=2}^4\kappa_j sI_{aj}+\varepsilon,\qquad\varepsilon\sim N(0,\sigma^2).
$$
The star in the formula includes both main effects and their <statistical interaction>. There are twelve coefficients: an intercept, four year contrasts, one sex contrast, three age contrasts and three sex-by-age contrasts. Year effects are assumed common across all sex-age combinations; there is no year <interaction term>. Independent errors with common <variance> are additional assumptions of the reported exact tests.
The <least-squares estimator> is $\widehat b=(X^TX)^{-1}X^TP$, where $X$ has rank twelve. Under the <normal linear model>,
$$
\widehat b\sim N_{12}(b,\sigma^2(X^TX)^{-1}),\qquad \frac{\mathrm{RSS}}{\sigma^2}\sim\chi^2_{28},
$$
and these quantities are independent. This standard normal-projection theorem gives $s^2=\mathrm{RSS}/28$ and estimated coefficient <standard errors> $s\sqrt{(X^TX)^{-1}_{jj}}$. A coefficient estimate divided by its <standard error> has a <Student t-distribution> with 28 <degrees of freedom> under its zero-coefficient null. These are the printed t statistics and two-sided <p-values>; a printed zero is rounding, not a probability of exactly zero. The option suppresses the coefficient-correlation display.
The fitted intercept $37.275$ is the expected percentage for the reference sex, age and year. Relative to that year, the year changes are $0.375,0.500,1.750,3.000$ percentage points. The last two are individually significant at the conventional five-percent level; the first two are not. The sex coefficient $-18.8$ is the female-minus-male difference in the youngest group. The male age changes are $-7,-12.8,-23$ points. Adding the three <interaction term> coefficients gives the female age changes $-5.2,-8.2,-13.8$. Thus the sex gap shrinks with age. An individual insignificant <interaction term> coefficient does not justify deleting the whole interaction: the joint <partial F-test> for the three interaction terms gives $F=18.6822$ on $(3,28)$ <degrees of freedom>, with $p\simeq7.46\times10^{-7}$.
The <regression residuals> are observed minus fitted percentages; their printed five-number summary describes their spread. Direct calculation gives $\mathrm{RSS}=60.2$, so the <residual standard error> is
$$
s=\sqrt{60.2/28}=1.46629\text{ percentage points}.
$$
The <coefficient of determination> is $1-\mathrm{RSS}/\mathrm{TSS}=0.985827$. The overall <F-test> compares all eleven non-intercept coefficients with zero:
$$
F=\frac{(\mathrm{TSS}-\mathrm{RSS})/11}{\mathrm{RSS}/28}=177.053.
$$
Under the null it has an <F-distribution> on $(11,28)$ <degrees of freedom>; its very small <p-value> shows substantial explained variation. This does not independently validate the Gaussian error assumptions. Survey percentages can have different sampling <variances> and correlations, and the survey denominators and design are not supplied. The output's exact inferential interpretation is conditional on the stated <normal linear model>.
The final command makes an <interaction plot>, not another regression fit. Its points are arithmetic averages over the five years for each sex-age combination. \b[It draws two decreasing, nonparallel lines], with coordinates
$$
\boxed{\text{men: }(38.4,31.4,25.6,15.4),\qquad\text{women: }(19.6,14.4,11.4,5.8).}
$$
Their gaps are $18.8,17.0,14.2,9.6$ percentage points, illustrating the sex-by-age <statistical interaction>. The average uses the software vector's disputed entry, consistently with the displayed fit. These are means of percentages, not denominator-weighted pooled prevalence estimates.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-38-interaction.png]
{title=Sex-by-age interaction plot averaged over years, using the printed software input}
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