2 Natasha is investigating binomial distributions. She constructs the spreadsheet in Fig. 2 which shows the first 3 and last 4 rows of a simulation involving two independent variables, \(X\) and \(Y\), with distributions \(\mathrm { B } ( 10,0.3 )\) and \(\mathrm { B } ( 50,0.3 )\) respectively. The spreadsheet also shows the corresponding value of the random variable \(Z\), defined by \(Z = 5 X - Y\), for each pair of values of \(X\) and \(Y\).
There are 100 simulated values of each of \(X , Y\) and \(Z\). The spreadsheet also shows whether each value of \(Z\) is greater than 6, and cells D103 and D104 show the number of values of \(Z\) which are greater than 6 and not greater than 6 respectively.
\begin{table}[h]
| 1 | A | B | C | D | E |
| 1 | X | Y | \(\mathbf { Z } = \mathbf { 5 } \mathbf { X } - \mathbf { Y }\) | \(\mathbf { Z } > \mathbf { 6 }\) | |
| 2 | 4 | 13 | 7 | Y | |
| 3 | 4 | 17 | 3 | N | |
| 4 | 3 | 21 | -6 | N | |
| 5 | | | | | |
| 6 | | | | | |
| 98 | 1 | 14 | -9 | N | |
| 99 | 5 | 12 | 13 | Y | |
| 100 | 3 | 18 | -3 | N | |
| 101 | 3 | 15 | 0 | N | |
| 102 | | | | | |
| 103 | | | Number of Y | 19 | |
| 104 | | | Number of N | 81 | |
| 105 | | | | | |
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{table}
- Use the information in the spreadsheet to write down an estimate of \(\mathrm { P } ( Z > 6 )\).
- Explain how a more reliable estimate of \(\mathrm { P } ( Z > 6 )\) could be obtained.
- State the greatest possible value of \(Z\).
- Explain why it is very unlikely that \(Z\) would have this value.
- Use the Central Limit Theorem to calculate an estimate of the probability that the mean of 20 independent values of \(Z\) is greater than 2 .