6 When Monica walks to work from home, she uses either route A or route B.
- Her journey time, \(X\) minutes, by route A may be assumed to be normally distributed with a mean of 37 and a standard deviation of 8 .
Determine:
- \(\mathrm { P } ( X < 45 )\);
- \(\mathrm { P } ( 30 < X < 45 )\).
- Her journey time, \(Y\) minutes, by route B may be assumed to be normally distributed with a mean of 40 and a standard deviation of \(\sigma\).
Given that \(\mathrm { P } ( Y > 45 ) = 0.12\), calculate the value of \(\sigma\).
- If Monica leaves home at 8.15 am to walk to work hoping to arrive by 9.00 am , state, with a reason, which route she should take.
- When Monica travels to work from home by car, her journey time, \(W\) minutes, has a mean of 18 and a standard deviation of 12 .
Estimate the probability that, for a random sample of 36 journeys to work from home by car, Monica's mean time is more than 20 minutes.
- Indicate where, if anywhere, in this question you needed to make use of the Central Limit Theorem.