Reverse-engineering from given variance

6 A random sample of 5 values of a variable \(X\) is given below. $$\begin{array} { l l l l l } 2 & 3 & 3 & 5 & a \end{array}$$

1. Find an expression, in terms of \(a\), for the mean of these values.
   It is given that an unbiased estimate of the population variance of \(X\), using these values, is 4 . It is also given that \(a\) is positive.
2. Find and simplify a quadratic equation in terms of \(a\) and hence find the value of \(a\).

6 A sample of 5 randomly selected values of a variable \(X\) is as follows: $$\begin{array} { l l l l l } 1 & 2 & 6 & 1 & a \end{array}$$ where \(a > 0\).
Given that an unbiased estimate of the variance of \(X\) calculated from this sample is \(\frac { 11 } { 2 }\), find the value of \(a\).

2 The length of worms is denoted by \(X \mathrm {~cm}\). The lengths of a random sample of 50 worms were measured. Some of the results were lost, but the following results are available.

• \(\Sigma x ^ { 2 } = 4361\)
• An unbiased estimate of the population variance of \(X\) is 9.62.

Calculate the mean length of the 50 worms.