3 The volume, \(X\) litres, of orange juice in a 1-litre carton may be modelled by a normal distribution with unknown mean \(\mu\). The volumes, \(x\) litres, recorded to the nearest 0.01 litre, in a random sample of 100 cartons are shown in the table.

1. For the group ' \(0.98 - 1.00\) ':
   1. show that it has a mid-point of 0.99 litres;
   2. state the minimum and the maximum values of \(x\) that could be included in this group.
2. Calculate, to three decimal places, estimates of the mean and the standard deviation of these 100 volumes.
3. 1. Construct an approximate \(99 \%\) confidence interval for \(\mu\).
   2. State why use of the Central Limit Theorem was not required when calculating this confidence interval.
   3. Give a reason why the confidence interval is approximate rather than exact.
4. Give a reason in support of the claim that:
   1. \(\mu > 1\);
   2. \(\mathrm { P } ( 0.94 < X < 1.16 )\) is approximately 1 .
      \includegraphics[max width=\textwidth, alt={}]{156f9453-ebc6-4406-b5bc-08d1918ebc62-10_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{156f9453-ebc6-4406-b5bc-08d1918ebc62-11_2486_1714_221_153}