Confidence intervals, which formula to use

[wtmoore]

I have created 1000 random samples of size 10, where X~N(8,2). For each sample I have calculated the sample mean, sample variance, (sigma₁)² and (sigma₂)².

I want to find a 95% confidence interval for the mean assuming the population variance is known, and a 95% confidence interval for the mean assuming the population variance is unknown.

What formula do I use for these?

I think, given that I am sampling from a normal distribution, for the known confidence interval, would be:

(X₁bar X₂bar) +- z_a/2s.e.(X₁bar-X₂bar) given that the samples are sufficiently small enough.

For unknown, I think it's:

(X1bar X2bar) +- t_{((n1+n2-2),a/2)} s.e(X₁bar-X₂bar) <<s.e has a tilda over it, also this formula assumes the sigmas are the same.

Is this correct?

 

[Stephen Tashi]

I have created 1000 random samples of size 10, where X~N(8,2). For each sample I have calculated the sample mean, sample variance, (sigma₁)² and (sigma₂)².

I want to find a 95% confidence interval for the mean assuming the population variance is known, and a 95% confidence interval for the mean assuming the population variance is unknown.

I don't understand why your notation involves two different means and variances when you have one population N(8,2) and 1000 samples of 10 realizations each. Where does the 1,2 come from? Are they the lower and upper bounds of the confidence interval?

(You should look at https://www.physicsforums.com/showthread.php?t=546968 instead of using the SUB tag.)

Can you explain your objective in doing this calculation. As I understand your post, you already know the population mean and variance. So why are you seeking a confidence interval for it? Is this an experiment in testing the theory of confidence intervals?

 

[wtmoore]

I don't understand why your notation involves two different means and variances when you have one population N(8,2) and 1000 samples of 10 realizations each. Where does the 1,2 come from? Are they the lower and upper bounds of the confidence interval?

(You should look at https://www.physicsforums.com/showthread.php?t=546968 instead of using the SUB tag.)

Can you explain your objective in doing this calculation. As I understand your post, you already know the population mean and variance. So why are you seeking a confidence interval for it? Is this an experiment in testing the theory of confidence intervals?

It's a tutorial question for a class I'm taking. It's basically revision, I will write out the formulas for the two sigmas. My understanding was they they were the population variances, but, I meant to say that the sigmas have hats on them for estimated, so I think they are sample variances that are estimates for the population.

both sigmas have a hat on for estimated
σ₁² = [sum_{(from i=1 to n)}(x_i-xbar)²]/n

σ₂² = same but divided by n+1(Will have a look at LaTeX, usually I get some of it to work and when it comes to summations and things I can't grasp it, which is weird as I'm quite good at coding in R, maple, etc)

 

[Stephen Tashi]

It's a tutorial question for a class I'm taking. It's basically revision

Revison? - do you mean "review"?

I think they are sample variances that are estimates for the population.

The formulas are estimators of the population variance. They are also two alternative definitions for "sample variance". Textbooks differ as to which definition they use.

That clears up the definition of the sigmas, but your statement of the assignment is still incoherent. For an assigned problem, its best to obey the format of the homework section of the forum and state the exercise exactly as it was assigned. Can you do that?

 

[wtmoore]

I can, but it's not homework, it is revision for exams, they are extra problems we can choose to do which will help us in exams.

The question is stated as it is, but instead of saying I wish to find, it says, find, I haven't left any information out, and have asked it in the first person instead.