In my attempt to understand the method to find a confidence interval for a population mean, given the population standard deviation, I am having trouble following the logic here. Please read, I have my questions embedded along the way of explaining the method.
First pick the Confidence Level (CL) (CL as decimal or CL% as a percentage). This is the level of confidence we possess that the population mean will be inside the confidence interval. First of all, what does the CL actually represent, a probability of what event?
Then find Alpha: α = 1 − CL
Then the next step is to find the Critical Probability (CP, p*). This is done: p* = 1 − α/2.
But doesnt this depend on one or two tails for the distribution?
Then find the Critical Value (CV). This is done by either the inverse Normal or the inverse T distributions. CV = N⁻¹(p*) or CV = T⁻¹(p*, df)
But is this right for the normal distribution option? or both? I figured I should put the actual CL in the inverse function, not the critical probability p*. Because N(95%) is covered by 95% of the area... not by p* = 97.5% of the area. Am I missing something here? The z-score table shows 0.95 as the portion of the distribution Im looking for, but should I be looking for 0.975 if I want a 95%?
Then find the Margin for Error (ME). ME = CV*SE
Where SE is the standard error of the mean at SE = s/√(n)
Confidence Interval (CI) = x̅ ± ME
Where am I going wrong in my concept?
Update:Will... I appreciate your response. Could you also tell me how the equations change for two tail or one tail?
Answers & Comments
Verified answer
first question
the cl is actually a measure in probability that your confident the population mean will be within this interval. meaning if 95% cl, i will be 95% sure that this population mean is truly in this interval. the event happens to be that the true population mean is in this interval.
second question.
yes this does depend on one or two tail. if it is two tail, your tails will have half the probability compared to a one tail test.
the middle questions.
youre not missing anything that i am knowing of. i like the fact that you know the difference between t and z, that is a very distinct difference. when you have a two tailed test, yes you should be looking for that 97.5 because the 5% is split into both of the sides due to symmetry giving you 2.5% in each tail. one tail test i would stick with the 95, two 97.5.
the last question.
i don't know what you mean by the last question. you last portion is correct. you put into the most simple terms you can frankly. just remember in the end when you do have the confidence interval, dont interpret it in the wrong way. it is just a way of saying directly i am ....% or 95% or whatever% confident that the true population mean lies in the interval of blah to blah.
The confidence interval for a population mean will have some probability that it is actually higher and some probability that it will actually be lower. By convention these are always taken to be equal. Therefore there are always two tails that each contain alpha/2 probability.
For example, suppose several cars of the same type are tested to find the average miles per gallon for that car. The 95% confidence interval for the mean may be 35 to 39 mpg. This means 2.5% probability that it is really under 35 mpg and 2.5% probability that it was really over 39 mpg. You wouldn't quote a one-tailed confidence interval that was open-ended, say at least 36 mpg having 5% probability that it was less.