# Finding or estimating confidence interval for populaion mean

1. Aug 25, 2011

### Rasalhague

From Koosis, I pieced together the following algorithm.

Is sigma known?

Yes? Then calculate the exact confidence interval using a normal distribution to estimate that of the sample means, with mean = the mean of sample means = the mean of the population, $\mu_{\overline{x}}=\mu$, and standard deviation $\sigma_{\overline{x}}=\sigma/\sqrt{n}$.

No? Then is the poplation normal?

Yes? Then (a) estimate the confidence interval with a Student's t distribution for the sample means, using degrees of freedom dof = n - 1, and standard deviation $s_{\overline{x}}=s\sqrt{n}$, or (b) for a slightly inferior result, and only if $n\geq 30$, estimate the confidence interval using the normal distribution with mean $\mu_{\overline{x}}=\mu$, and standard deviation $s_{\overline{x}}=s\sqrt{n}$.

No or don't know? Then is $n\geq 30$?

Yes? Then estimate the confidence interval, using a normal distribution to estimate that of the sample means, with mean $\mu_{\overline{x}}=\mu$, and standard deviation $s_{\overline{x}}=s\sqrt{n}$.

No? Then can't do.

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But Sanders has the following, somewhat different algorithm.

Is $n\geq 30$?

Yes? Then use z values in computations.

No? Then are population values known to be normally distributed?

Yes? If the population standard deviation of the population is known, use z values in computations. Otherwise, use t values in computations.

No? Cannot use z or t values in computations.

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Any comments on which is the best procedure? Actually Koosis presented the z test first, as if he, like Sanders, assumed that one would choose this over the t test wherever possible, even though he said it wasn't as good when both choices were possible. I wonder why z beats t in that case? Is it because the difference in accuracy is negligible then and the computations for t potentially more time consuming than those for z? (And if so, is this still the case with current software; both books are a few years old.)

2. Aug 27, 2011

### I like Serena

Hi Rasalhague!