Given a population sample..

(X_Bar - mu)/(S/sqrt(n)) ~ t distribution with n-1 degrees of freedom

this fact is used to derive the (1-alpha)100% confidence interval for population mean(mu) which is..

X_Bar +/- (S/sqrt(n))t_n-1_1-alpha/2

where t_n-1_1-alpha/2 is (1-alpha/2)th quantile of t-distribution with n-1 degrees of freedom.

Note that this interval is random that is if you take multiple samples with same sample size then the value of this interval will most likely be different for different samples from the same population. However, the true population mean(mu) has a fixed value.

One interpretation of confidence interval is that if you take multiple samples and find some confidence interval, say 95% confidence interval, from all the samples then 95% of the time mu will lie inside those intervals.

(X_Bar - mu)/(S/sqrt(n)) ~ t distribution with n-1 degrees of freedom

this fact is used to derive the (1-alpha)100% confidence interval for population mean(mu) which is..

X_Bar +/- (S/sqrt(n))t_n-1_1-alpha/2

where t_n-1_1-alpha/2 is (1-alpha/2)th quantile of t-distribution with n-1 degrees of freedom.

Note that this interval is random that is if you take multiple samples with same sample size then the value of this interval will most likely be different for different samples from the same population. However, the true population mean(mu) has a fixed value.

One interpretation of confidence interval is that if you take multiple samples and find some confidence interval, say 95% confidence interval, from all the samples then 95% of the time mu will lie inside those intervals.

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