Dear specialists, I would appreciate your support in this matter. I understand It is reasonable to conclude that sampling spread (for a normal sampling distribution) does affect mean confidence interval (sampling spread is a population spread point estimator and the formulae for CI presents standard deviation). However, I am not quite sure if this is true for all possible population parameters, or how to prove it. Could you please help me learn more?
Does population spread affect all possible population parameters confidence interval?
- Thread starter Gustavo Borges
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Willing to engage in this dialog.
CI = +/- (z-value * (Std / sqrt (n)) or
CI = +/- (z-value * (SE), which SE is considered the population std
So if we assume z-value is just a coverage value (e.g., 1.96) we can consider it a constant, then we get:
CI = estimate +/- constant * population standard deviation.
The population standard deviation gets smaller and smaller as the sample size increases.
What are your thoughts on these statements. A simulation may help quantify what the value may converge to as the sample approaches the population size.
CI = +/- (z-value * (Std / sqrt (n)) or
CI = +/- (z-value * (SE), which SE is considered the population std
So if we assume z-value is just a coverage value (e.g., 1.96) we can consider it a constant, then we get:
CI = estimate +/- constant * population standard deviation.
The population standard deviation gets smaller and smaller as the sample size increases.
What are your thoughts on these statements. A simulation may help quantify what the value may converge to as the sample approaches the population size.
Thanks for your answer, and sorry for the confusion. As I understand, we select samples from a population to estimate a population parameter (i.e. mean, median, variance, etc). From these selected samples, we can estimate a population parameter and a confidence interval for such estimation. This confidence interval width (range) will depend on the number of samples we have collected from the population (sample size) and our significance level (i.e.: CI 95%, CI 99%, etc.). My question is: Apart from the estimations for mean and median, does the population spread (variance) affect the width of the confidence interval? (is that true for all possible population estimators, regardless the sampling distribution?)
Depends on what you mean by "population variance" and which statistical model you are referring to.
Simple example. Say you are asked to compute a 95% C.I. for the regression coefficient in the simple bivariate regression model \(Y = \hat{\beta_{0}} + \hat{\beta_{1}}X + \hat{\epsilon}\). By assumption, \(\epsilon \sim N(0, \sigma^2_{\epsilon})\). So if by "population variance" you mean \(\sigma^2_{\epsilon}\) then yes. But if by "population variance" you mean, say, the variance of your predictor \(X\), then the answer is "no".
As @hlsmith said, your question, as phrased, is ambiguous. If you do not define what you mean by (a) population variance and (b) the model you refer to, there is no way to answer this.
Simple example. Say you are asked to compute a 95% C.I. for the regression coefficient in the simple bivariate regression model \(Y = \hat{\beta_{0}} + \hat{\beta_{1}}X + \hat{\epsilon}\). By assumption, \(\epsilon \sim N(0, \sigma^2_{\epsilon})\). So if by "population variance" you mean \(\sigma^2_{\epsilon}\) then yes. But if by "population variance" you mean, say, the variance of your predictor \(X\), then the answer is "no".
As @hlsmith said, your question, as phrased, is ambiguous. If you do not define what you mean by (a) population variance and (b) the model you refer to, there is no way to answer this.
Thank you for your support, some common ground in nomenclature it may help here.
Population: All possible items in a study / analysis. (i.e.: All farm sheep in Ireland )
Population parameter: A given parameter from that population (i.e.: Average number of sheep per farm). That is a number and has no confidence interval (true value).
Sampling: A subset of this population to be used to infer on an population parameter (i.e. 10% of all farms). Data coming from sampling may fit in any probability distribution and I would expect it to fit it in a normal distribution.
Population estimator: A estimation of a population parameter based on the sampling data. Because it is a estimation, it is presented as a expected value and a range (CI) confidence interval (i.e.: Average is 100 sheep per farm with CI95% equal to 90 and 110; in other words 100 +/- 10 sheep per farm).
I understand that the population spread /variance (what is a population parameter) affects the confidence interval of mean (population estimator) if my sampling fits in a normal distribution. This is because the sampling variance appears on the formula for normally distributed samplings mean CI; furthermore, sampling variance is also is a population estimator for the population spread / variance for normally distributed samplings. However, I do not know if the relationship between population spread/variance and confidence interval is valid for all possible population parameters estimators and for all possible sampling distributions. Is there a common rule? Is there a subset where there is a known relationship?
For instance, I could be interested on estimating the population standard deviation with a sampling that follows a hypergeometric distribution, In that case, would the population spread affect population estimator CI?
I hope it is clearer now. I do appreciate you taking your time to easy my curiosity; I do apologize for not making this clearer before.
Population: All possible items in a study / analysis. (i.e.: All farm sheep in Ireland )
Population parameter: A given parameter from that population (i.e.: Average number of sheep per farm). That is a number and has no confidence interval (true value).
Sampling: A subset of this population to be used to infer on an population parameter (i.e. 10% of all farms). Data coming from sampling may fit in any probability distribution and I would expect it to fit it in a normal distribution.
Population estimator: A estimation of a population parameter based on the sampling data. Because it is a estimation, it is presented as a expected value and a range (CI) confidence interval (i.e.: Average is 100 sheep per farm with CI95% equal to 90 and 110; in other words 100 +/- 10 sheep per farm).
I understand that the population spread /variance (what is a population parameter) affects the confidence interval of mean (population estimator) if my sampling fits in a normal distribution. This is because the sampling variance appears on the formula for normally distributed samplings mean CI; furthermore, sampling variance is also is a population estimator for the population spread / variance for normally distributed samplings. However, I do not know if the relationship between population spread/variance and confidence interval is valid for all possible population parameters estimators and for all possible sampling distributions. Is there a common rule? Is there a subset where there is a known relationship?
For instance, I could be interested on estimating the population standard deviation with a sampling that follows a hypergeometric distribution, In that case, would the population spread affect population estimator CI?
I hope it is clearer now. I do appreciate you taking your time to easy my curiosity; I do apologize for not making this clearer before.
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