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# Confidence Intervals watch

1. Can someone please explain to me the theory behind confidence intervals (for Stats)? I understand how you use the formulae.. but thats about it. Any help would be great!
2. Confidence Intervals are weird...like if you have a 95% interval, it means that 95 out of 100 times, the sample mean will lie in that region. e.g. If you have a population (e.g. height of people), and you take 100 samples from that: then with a 95%~[156, 190] means that 95 of the means taken from the population will lie between 156 and 190...kinda hard to explain. Alot of the stuff in S3 is just madness lol, like the chi-square distribution for binomial and poisson...its such a pain to work out!
3. (Original post by kenkennykenken)
Confidence Intervals are weird...like if you have a 95% interval, it means that 95 out of 100 times, the sample mean will lie in that region. e.g. If you have a population (e.g. height of people), and you take 100 samples from that: then with a 95%~[156, 190] means that 95 of the means taken from the population will lie between 156 and 190...kinda hard to explain. Alot of the stuff in S3 is just madness lol, like the chi-square distribution for binomial and poisson...its such a pain to work out!
Thanks for trying to help. Do you know where the Central Limit Theorem fits into all this?
4. we are doing these right now! and we're doing them central theorem thingymujiggs next week!!

how lovely and easy and enjoyable is stats compared to pure!!!???

it's bliss!
5. (Original post by sparkler)
we are doing these right now! and we're doing them central theorem thingymujiggs next week!!

how lovely and easy and enjoyable is stats compared to pure!!!???

it's bliss!
seriously!?! I find Pure much easier. Stats is a nightmare.

So can you explain the theory behind it? Or is there a good section in your text book that explains it well? My books are crap..
6. (Original post by Elle)
Can someone please explain to me the theory behind confidence intervals (for Stats)? I understand how you use the formulae.. but thats about it. Any help would be great!
Elle,

A ß % confidence interval, suggests that, when repeated (infinitely) many times, the population mean µ will lie in ß % of the resulting intervals.

(Original post by Elle)
Do you know where the Central Limit Theorem fits into all this?
Confidence intervals are formed by:

Sample Mean ± k*Standard Error

Where k is some quantile, eg for 95 % C.I, we use PHI^-1(0.975)=1.96

It is the Standard Error (sigma/sqrt n), which is vital here:

The Central Limit Theorum (C.L.T) states that when n is large, and getting larger, the distribution of the Sample Means (X-Bar) is NORMAL.

X-Bar ~ N (µ, sigma^2 / n).

Hopefully now you can see the link.
7. (Original post by Expression)
Elle,

A ß % confidence interval, suggests that, when repeated (infinitely) many times, the population mean µ will lie in ß % of the resulting intervals.

Confidence intervals are formed by:

Sample Mean ± k*Standard Error

Where k is some quantile, eg for 95 % C.I, we use PHI^-1(0.975)=1.96

It is the Standard Error (sigma/sqrt n), which is vital here:

The Central Limit Theorum (C.L.T) states that when n is large, and getting larger, the distribution of the Sample Means (X-Bar) is NORMAL.

X-Bar ~ N (µ, sigma^2 / n).

Hopefully now you can see the link.
Thanks Expression!!

I kind of understand it, but I'm still a little unclear about the C.L.T. Is the symmetry of a Confidence Interval created by the C.L.T because it's a normal distribution?
8. (Original post by Expression)
eg for 95 % C.I, we use PHI^-1(0.975)=1.96
don't you just use stats tables???

like for a 95% C.I. its the 'U of 0.025'

hmmph..
9. (Original post by sparkler)
we are doing these right now! and we're doing them central theorem thingymujiggs next week!!

how lovely and easy and enjoyable is stats compared to pure!!!???

it's bliss!
Yea we r doing all that in S3 rite now too.. abt the central limit theorem all i know is, u can use to too using the normal distribution with any other kinda dist..

ooo yeaa i love stats compared too pure esp compared 2 P3, ITS hell, i have a differentiation test tommorow its awful coz the CL matches r on tonite, both milan & arsenal and i will sleep at 1:00 am my time, i m gonno sleep all over my test.. btw i m trying 2 find S3 papers.. anyone remmeber mr hughes username and password?? help!
10. I got all the past S3 papers if you want them. Ohh i actually prefer pure, coz its more straight forward, rather than having to spend most of the exam interpreting the paper like in stats!
11. (Original post by Elle)
Thanks Expression!!

I kind of understand it, but I'm still a little unclear about the C.L.T. Is the symmetry of a Confidence Interval created by the C.L.T because it's a normal distribution?
Right, I'll quote exactly what I was told at A-Level S1 (picks up big red box full of A Level orange exercise books)

A. Sampling Distribution of the Mean

The distribution of the sample means is approximately normal.

The mean of the distribution of the sample means is equal to the mean of the parent population.

The larger the sample size, n, the more tightly clustered around its mean is the distribution. (ie the varience of the distribution of the sample means is smaller).

If population variance is sigma² and samples are of size n, then distribution of sample means of such samples have variance:
sigma²/ n.

Population:
Mean = µ
Variance = sigma²

Sample:
Sample Size = n
Mean = µ
Variance = sigma²/n

The Central Limit Theorum states that even if the parent population is not normal then the following is true:

1. the distribution of the sample means will be NORMAL if the sample size is large enough
2. the mean of the sample is approximately equal to the mean of the parent population
3. the variance of the distribution is appoximately the variance of the parent population divided by the size of the sample
4. approximations get closer to normal distribution as sample size gets bigger
5. a good rule of thumb is n >= 30

Example - Use of CLT in Normal Distribution problems
The mean weight of trout in a fish farm is 980g and the standard deviation is 100g.

What is the probability that a catch of 10 trout will have a mean weight per fish of 1050g ?

Solution
Assuming weight of the trout, W, is normally distributed, then parent population:
W~N(980, 100²)

The catch of the fish was of size 10, hense n=10.

By the Central Limit Theorum the mean weight of the samples of size 10:
• is NORMALLY DISTRIBUTED
• has mean of 980g
• has a variance of 100²/10 = 1000

or W-bar ~ N(980, 1000)

Standardizing the score

z= (x - x-bar)/sqrt(Var)
= (1050 - 980)/sqrt(1000)
= 2.21

P(W-bar > 1050) = P(z>2.21) = 1-P(z<=2.21) = 1-PHI(2.21) = 1-0.9864 = 0.0136.

So there is a 1.36 % chance of all samples of size 10, having a mean weight per fish of more than 1050g.
12. (Original post by kenkennykenken)
I got all the past S3 papers if you want them. Ohh i actually prefer pure, coz its more straight forward, rather than having to spend most of the exam interpreting the paper like in stats!
Wow can u attach them via email pleasE? my email id i will pm to u yea? is that ok.. but what abt the solutions.. i kinda need them too...
13. (Original post by sparkler)
don't you just use stats tables???

like for a 95% C.I. its the 'U of 0.025'

hmmph..
Your "U" function must be something that you have which works based on a area to the right of a score z.

Where as "PHI", seen in most tables, works on areas to the left of a score z.

PHI^-1 therefore turns a known area (0.975) lying to the left of a now desired score z, into that score, and I imagine that your "U" function performs this operation just on the other side, since PHI^-1 (0.975) =?= U(0.025)
14. Thanks!! Thats a HUGE help!!!
15. (Original post by Expression)
Your "U" function must be something that you have which works based on a area to the right of a score z.

Where as "PHI", seen in most tables, works on areas to the left of a score z.

PHI^-1 therefore turns a known area (0.975) lying to the left of a now desired score z, into that score, and I imagine that your "U" function performs this operation just on the other side, since PHI^-1 (0.975) =?= U(0.025)
well now, that's just weird!!!

16. I have another question about confidence intervals (but I didn't really want to start a new thread) and any help would be great!

If I calculate a 75 per cent confidence interval for a sample of data, and then also calculate it's 95 per cent confidence interval.. how would I go about comparing and analysing the difference between them?

17. (Original post by Elle)
I have another question about confidence intervals (but I didn't really want to start a new thread) and any help would be great!

If I calculate a 75 per cent confidence interval for a sample of data, and then also calculate it's 95 per cent confidence interval.. how would I go about comparing and analysing the difference between them?

Obviously the 95% confidence interval would be greater if I remember correctly. But you can only compare the confidence intervals of different data sets. In an exam you may be given a value and be asked that according to the confidence interval you calculated whether or not that value would be a good estimate of the mean. Obviously if the value is within the confidence interval then the value is a viable estimate of the mean.
18. (Original post by ZJuwelH)
Obviously the 95% confidence interval would be greater if I remember correctly. But you can only compare the confidence intervals of different data sets. In an exam you may be given a value and be asked that according to the confidence interval you calculated whether or not that value would be a good estimate of the mean. Obviously if the value is within the confidence interval then the value is a viable estimate of the mean.
Thanks Juwel.. but my teacher suggested that I compare different levels of confidence for one data set (for coursework).. not quite sure how he wants me to do that.. hmm..
19. (Original post by Elle)
Thanks Juwel.. but my teacher suggested that I compare different levels of confidence for one data set (for coursework).. not quite sure how he wants me to do that.. hmm..
In our coursework we did the 90% and 95% intervals for each of our three data sets. It was easy for us to compare across the data sets, especially if you did a graphical representation. But I have no idea how you'd compare different CI for the same data set.

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