5 Layla works at an internet café. Each terminal at the café has its own keyboard, and keyboards need to be replaced whenever faults develop.
Layla knows that the number of weeks for which a keyboard lasts before it needs to be replaced can be modelled by the random variable \(X\), which has an exponential distribution with mean 20 and variance 400 . She wants to investigate how likely it is that the keyboard at a terminal will need to be replaced at least 3 times within a year (taken as being a period of 52 weeks).
Layla designs the simulation shown in the spreadsheet below. Each of the 20 rows below the heading row consists of 3 values of \(X\) together with their sum \(T\). All of the values in the spreadsheet have been rounded to 1 decimal place.
| A | B | C | D |
| 1 | \(\mathrm { X } _ { 1 }\) | \(\mathrm { X } _ { 2 }\) | \(\mathrm { X } _ { 3 }\) | T |
| 2 | 10.9 | 21.5 | 5.3 | 37.7 |
| 3 | 23.9 | 52.4 | 85.3 | 161.6 |
| 4 | 5.2 | 10.4 | 24.0 | 39.6 |
| 5 | 2.9 | 14.4 | 0.8 | 18.1 |
| 6 | 9.0 | 43.3 | 49.7 | 102.0 |
| 7 | 0.4 | 16.2 | 12.4 | 29.0 |
| 8 | 44.1 | 39.5 | 22.1 | 105.7 |
| 9 | 9.2 | 43.6 | 13.9 | 66.7 |
| 10 | 40.4 | 10.9 | 6.1 | 57.4 |
| 11 | 3.2 | 54.8 | 15.7 | 73.7 |
| 12 | 5.3 | 6.1 | 1.6 | 13.0 |
| 13 | 20.5 | 28.9 | 22.9 | 72.3 |
| 14 | 37.3 | 2.1 | 28.6 | 68.0 |
| 15 | 7.1 | 13.6 | 50.1 | 70.8 |
| 16 | 18.6 | 2.0 | 9.3 | 29.9 |
| 17 | 9.0 | 1.2 | 49.9 | 60.1 |
| 18 | 1.9 | 9.5 | 69.8 | 81.2 |
| 19 | 9.0 | 2.1 | 10.4 | 21.5 |
| 20 | 28.7 | 1.4 | 93.8 | 123.9 |
| 21 | 1.8 | 2.9 | 34.8 | 39.5 |
- Explain why \(T\) represents the number of weeks after which the third keyboard at a terminal will need to be replaced.
- Use the information in the spreadsheet to write down an estimate of \(\mathrm { P } ( T > 52 )\).
- Explain how you could obtain a more reliable estimate of \(\mathrm { P } ( T > 52 )\).
- The internet café has 50 terminals. You are given that faults in keyboards occur independently of each other.
Determine an estimate of the probability that the mean number of weeks before which the third keyboard at a terminal needs to be replaced is more than 52 .