1. ## Uniform Distribution question

Hey all, this is from a past exam but we were not given any solutions.

The Question:
Many students find that sitting through an entire lecture is difficult to do without falling asleep. Suppose the number of minutes a single, randomly selected student is asleep during a lecture is uniformly distributed between 7 and 14 minutes.

1.) Over the course of the semester (say, 50 lectures), what is the probability that a randomly selected student will sleep a total of between 130 and 140 minutes?
Assume that the minutes slept in any lecture is independent of the number of minutes slept in any other lectures.

2.) How many lectures must a student attend to be 95% sure that they will have slept at least 28 minutes in total?

I have done previous parts to the question that ask for E(X) and Std Deviation but I can't work out these 2.

Any help would be appreciated.

Thanks

2. Originally Posted by Nguyen
Hey all, this is from a past exam but we were not given any solutions.

The Question:
Many students find that sitting through an entire lecture is difficult to do without falling asleep. Suppose the number of minutes a single, randomly selected student is asleep during a lecture is uniformly distributed between 7 and 14 minutes.

1.) Over the course of the semester (say, 50 lectures), what is the probability that a randomly selected student will sleep a total of between 130 and 140 minutes?
Assume that the minutes slept in any lecture is independent of the number of minutes slept in any other lectures.

2.) How many lectures must a student attend to be 95% sure that they will have slept at least 28 minutes in total?

I have done previous parts to the question that ask for E(X) and Std Deviation but I can't work out these 2.

Any help would be appreciated.

Thanks
I suggest using the Central Limit Theorem.

3. Originally Posted by mr fantastic
I suggest using the Central Limit Theorem.
Thanks mr fantastic, but I am getting large numbers and it's not working out for me. Could you show working?

Thanks.

4. Originally Posted by Nguyen
Thanks mr fantastic, but I am getting large numbers and it's not working out for me. Could you show working?

Thanks.
You need to show your working.

5. Originally Posted by mr fantastic
You need to show your working.
oh yes, of course.

1. $\displaystyle \bar{X} \sim N \left( \mu = \frac{21}{2}, \sigma^2 = \frac{7}{50 \cdot 12} \right)$

$\displaystyle P(\bar{X} >130) = P \left( \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} > \frac{130-(21/2)}{(\sqrt{21}/6)/\sqrt{50}} \right)$

Same goes for $\displaystyle P(\bar{X} <140)$.
And that is obviously wrong so I am definitely using the wrong values.

2. Don't know what to do for this one.

Any help would be nice.

Thanks

6. Originally Posted by Nguyen
oh yes, of course.

1. $\displaystyle \bar{X} \sim N \left( \mu = \frac{21}{2}, \sigma^2 = \frac{7}{50 \cdot 12} \right)$

$\displaystyle P(\bar{X} >130) = P \left( \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} > \frac{130-(21/2)}{(\sqrt{21}/6)/\sqrt{50}} \right)$

Same goes for $\displaystyle P(\bar{X} <140)$.
And that is obviously wrong so I am definitely using the wrong values.

2. Don't know what to do for this one.

Any help would be nice.

Thanks
The variance of the time spent sleeping in a lecture is $\displaystyle 7^2/12 \text{ min}^2$, so the variance of the time slept in a total of 50 lectures is $\displaystyle 50 (7^2/12)\approx (14.29 \text{ min})^2$.

And the mean times spent sleeping in 50 lectures is $\displaystyle 50(10.5) = 525 \text{ min}$, so obviously for (1) the answer will be indistinguishable from zero.

This suggests the possibility of a typo.

CB

7. Originally Posted by CaptainBlack
The variance of the time spent sleeping in a lecture is $\displaystyle 7^2/12 \text{ min}^2$, so the variance of the time slept in a total of 50 lectures is $\displaystyle 50 (7^2/12)\approx (14.29 \text{ min})^2$.

And the mean times spent sleeping in 50 lectures is $\displaystyle 50(10.5) = 525 \text{ min}$, so obviously for (1) the answer will be indistinguishable from zero.

This suggests the possibility of a typo.

CB
Thanks CaptainBlack.
That's the problem for me, it is not a typo unfortunately.
For 1. my tutor said it has to do with working with summations of a random variable and using what we know about how $\displaystyle \bar{X}$ is distributed. But it is not working.

8. Originally Posted by Nguyen
Thanks CaptainBlack.
That's the problem for me, it is not a typo unfortunately.
For 1. my tutor said it has to do with working with summations of a random variable and using what we know about how $\displaystyle \bar{X}$ is distributed. But it is not working.
The typo may be in the problem as set. What has been discussed does work with summations of a random variable and using what we know about how $\displaystyle \overline{X}$ is distributed. The typo is probably in the number of lectures in a semester, 15 would seem to make more sense of the problem.

CB

9. Hi Huyen Nguyen,

Maybe you shouldn't lie and just do the STAT1003 assignment yourself. Im fairly certain that ANU stipulates you must not get solutions from other people...including forums. In case you weren't aware, lectures are on monday-wednesday. Go to them and learn something instead of cheating.

10. This thread is under investigation.