5 Each evening Aaron sets his alarm for 7 am. He believes that the probability that he wakes before his alarm rings each morning is 0.4 , and is independent from morning to morning.
- Assuming that Aaron's belief is correct, determine the probability that, during a week (7 mornings), he wakes before his alarm rings:
- on 2 or fewer mornings;
- on more than 1 but fewer than 5 mornings.
- Assuming that Aaron's belief is correct, calculate the probability that, during a 4 -week period, he wakes before his alarm rings on exactly 7 mornings.
- Assuming that Aaron's belief is correct, calculate values for the mean and standard deviation of the number of mornings in a week when Aaron wakes before his alarm rings.
(2 marks) - During a 50-week period, Aaron records, each week, the number of mornings on which he wakes before his alarm rings. The results are as follows.
| Number of mornings | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Frequency | 10 | 8 | 7 | 7 | 5 | 5 | 4 | 4 |
- Calculate the mean and standard deviation of these data.
- State, giving reasons, whether your answers to part (d)(i) support Aaron's belief that the probability that he wakes before his alarm rings each morning is 0.4 , and is independent from morning to morning.
(3 marks)