confidence interval and sampling size accuracy confusion.

[SadieKhan]

Hi,
I have to find the best interpretation of a confidence interval and I am kind of stuck, i always explain my confidence interval something like,
We are X% confident that say popu;ation mean lies between this and that.
But there are 4 choices given I know I have to chose we are X% confident about this and that statement but all these 4 choices look the same to me, I dont know what is the flaw , can you help me?
thanks
Question
"A 90% confidence interval for the number of hours that full-time college students sleep during a weekday is lower bound: 7.8 hours and upper bound: 8.8 hours. Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw.
"


1."90% of full-time college students sleep between 7.8 hours and 8.8 hours.
"
2.We are 90% confident that the mean number of hours of sleep that full-time college students get any day of the week is between 7.8 hours and 8.8 hours.


3."There is 90% probability that the mean hours of sleep that full-time college students get during a weekday is between 7.8 hours and 8.8 hours.
"


4."We are 90% confident that the mean hours of sleep that full-time college students get during a weekday is between 7.8 hours and 8.8 hours.
"
i think option 4 looks reasonable, but I dont know why we cant say that there is 90% probability???
Any suggestion
The other question is
"What effect does doubling the required accuracy have on the sample size? Why is this the expected result?
"

What does required accuracy mean??? does it mean alpha? or e?

[SmoothJohn]

Dothe words "symmetric" or "symmetrical" show up somewhere in the original question?

[BGM]

Note the confidence interval you have has the form



Choice 1 has the obvious flaw. This confidence interval is obviously different from



I guess you may not have the same distribution for the number of hours for
sleep on the weekend, so Choice 2 also has the flaw.

Choice 3 points out the common error;
It is because now we are frequentist, we believe
the unknown parameter of interest is a deterministic
constant, but not a random variable. The probability of a deterministic
constant fall in a pre-specified interval is either 1 (true) or 0 (false).
You cannot assign any probability here.
We are using a random interval to cover the true parameter.

I guess the accuracy here means the length of your interval.

[SadieKhan]

I guess the accuracy here means the length of your interval.

I still couldnt get it .. you mean the difference between upper bound and lower bound ?

[BGM]

Yes thats my guess.
Doubling the accuracy should mean reduce the length by half.

[SadieKhan]

Got it now thanks.
Does it have to do anything with margin of error.??
The margin of error is a statistic expressing the amount of random sampling error in a survey's results. The larger the margin of error, the less faith one should have that the poll's reported results are close to the "true" figures; that is, the figures for the whole population.
Also
The margin of error E is half of the width of the confidence interval. so by doubling the width of a confidence interval we are doubling the margin of error too, no?
Then I can show some calculations that since e has doubled thus our sample size has increased or decreased?
thanks