\begin{enumerate}
  \item Two classes of students, class $A$ and class $B$, sat a test.
\end{enumerate}

Class $A$ has 10 students. Class $B$ has 15 students.

Each student achieved a score, $x$, on the test and their scores are summarised in the table below.

\begin{center}
\begin{tabular}{ | c | c | c | c | }
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & $n$ & $\sum x$ & $\sum x ^ { 2 }$ \\
\hline
Class $A$ & 10 & 770 & 59610 \\
\hline
Class $B$ & 15 & $t$ & 58035 \\
\hline
\end{tabular}
\end{center}

The mean score for Class $A$ is 77 and the mean score for Class $B$ is 61\\
(a) Find the value of $t$\\
(b) Calculate the variance of the test scores for each class.

The highest score on the test was 95 and the lowest score was 45

These were each scored by students from the same class.\\
(c) State, with a reason, which class you believe they were from.

The two classes are combined into one group of 25 students.\\
(d) (i) Find the mean test score for all 25 students.\\
(ii) Find the variance of the test scores for all 25 students.

The teacher of class $A$ later realises that he added up the test scores for his class incorrectly. Each student's test score in class $A$ should be increased by 3\\
(e) Without further calculations, state, with a reason, the effect this will have on\\
(i) the variance of the test scores for class $A$\\
(ii) the mean test score for all 25 students\\
(iii) the variance of the test scores for all 25 students.

\begin{center}

\end{center}

\hfill \mbox{\textit{Edexcel S1 2018 Q1 [12]}}