10 Sarah takes a bus to work each weekday morning and returns each evening. The times in minutes that she has to wait for the bus in the morning and evening are modelled by uniform distributions over the intervals \([ 0,10 ]\) and \([ 0,6 ]\) respectively. The times in minutes for the bus journeys in the morning and evening are modelled by \(\mathrm { N } ( 25,4 )\) and \(\mathrm { N } ( 28,16 )\) respectively. You should assume that all of the times are independent. The total time in minutes that she takes for her two journeys, including the waiting times, is denoted by the random variable \(T\).
The spreadsheet below shows the first 20 rows of a simulation of 500 return journeys. It also shows in column H the numbers of values of \(T\) that are less than or equal to the corresponding values in column G. For example, there are 156 out of the 500 simulated values of \(T\) which are less than or equal to 58 minutes. All of the times have been rounded to 2 decimal places.
| A | B | C | D | E | F | G | H |
| 1 | Waiting time morning | Journey time morning | Waiting time evening | Journey time evening | Total time | | Total time t | Number \(\leqslant \mathbf { t }\) |
| 2 | 0.89 | 20.78 | 1.88 | 26.30 | 49.86 | | 46 | 0 |
| 3 | 3.55 | 21.24 | 1.04 | 29.61 | 55.44 | | 48 | 4 |
| 4 | 2.13 | 21.83 | 2.40 | 28.64 | 55.00 | | 50 | 13 |
| 5 | 5.12 | 25.04 | 3.13 | 24.30 | 57.60 | | 52 | 35 |
| 6 | 4.03 | 27.49 | 2.19 | 30.81 | 64.52 | | 54 | 57 |
| 7 | 2.47 | 20.54 | 4.32 | 34.61 | 61.93 | | 56 | 104 |
| 8 | 3.21 | 26.93 | 3.78 | 27.66 | 61.58 | | 58 | 156 |
| 9 | 9.72 | 24.15 | 0.63 | 27.53 | 62.03 | | 60 | 218 |
| 10 | 1.59 | 28.45 | 0.08 | 35.87 | 65.99 | | 62 | 288 |
| 11 | 7.34 | 23.04 | 4.02 | 24.77 | 59.17 | | 64 | 357 |
| 12 | 1.04 | 24.69 | 1.66 | 31.95 | 59.33 | | 66 | 408 |
| 13 | 7.17 | 22.16 | 2.55 | 25.39 | 57.28 | | 68 | 441 |
| 14 | 5.20 | 26.97 | 2.41 | 30.05 | 64.62 | | 70 | 475 |
| 15 | 5.01 | 26.84 | 1.88 | 36.21 | 69.93 | | 72 | 490 |
| 16 | 3.76 | 26.03 | 2.21 | 30.96 | 62.96 | | 74 | 496 |
| 17 | 0.96 | 23.72 | 2.55 | 29.36 | 56.59 | | 76 | 500 |
| 18 | 8.64 | 24.97 | 2.82 | 26.39 | 62.82 | | | |
| 19 | 0.59 | 20.82 | 4.57 | 31.41 | 57.38 | | | |
| 20 | 9.85 | 23.68 | 5.54 | 29.92 | 68.99 | | | |
| 01 | | | | | | | | |
- Use the spreadsheet output to estimate each of the following.
- \(\mathrm { P } ( T \leqslant 56 )\)
- \(\mathrm { P } ( T > 61 )\)
- The random variable \(W\) is Normally distributed with the same mean and variance as \(T\). Find each of the following.
- \(\mathrm { P } ( W \leqslant 56 )\)
- \(\mathrm { P } ( W > 61 )\)
- Explain why, if many more journeys were used in the simulation, you would expect \(\mathrm { P } ( T > 61 )\) to be extremely close to \(\mathrm { P } ( W > 61 )\).