Confidence interval

A 95 per cent confident for a population mean looks like.

Failed to parse (unknown function "\huge"): {\displaystyle \huge \mu = {\bar x}\pm1.96\frac{\sigma}{\sqrt n}}

In practice we seldom know that $\sigma$ is known. Therefore most of the times an estimate of standard deviation in the sample $s$ (sometimes denoted as Failed to parse (unknown function "\Large"): {\displaystyle \Large\hat\sigma} ) is used:

Failed to parse (unknown function "\huge"): {\displaystyle \huge \mu = {\bar x}\pm z\frac{s}{\sqrt n}}

If we use: Failed to parse (unknown function "\huge"): {\displaystyle \huge e=z \frac {s}{\sqrt n}}

We can make a conclusion on determining the minimum sample size given a requested error level and confidence level:: Failed to parse (unknown function "\huge"): {\displaystyle \huge n=(\frac {zs}{e})^2}

Adding up confidence intervals for the difference of means is given by: Failed to parse (unknown function "\huge"): {\displaystyle \huge (\mu_1 - \mu_2) = (\bar{x_1} - \bar{x_2}) \pm \sqrt{(\frac{s_1^2}{n_1^2}+\frac{s_2^2}{n_2^2})}}

When the sample size comes closer to the population size, the error will become smaller. Using the finite population correction factor this can be taken into account:

Failed to parse (unknown function "\huge"): {\displaystyle \huge \sqrt {(1-\frac {n}{N})}}

So corrected with the finite population correction factor a 95 per cent confident for a population mean looks like: Failed to parse (unknown function "\huge"): {\displaystyle \huge \mu = {\bar x}\pm 1.96 \sqrt {(1-\frac {n}{N})} \frac{s}{\sqrt n}}

The T-distribution can be used instead of the Normal distribution when defining confidence intervals when sample size is small or the standard deviation is unknown.

Part of M7-Quantitative Decision Methods-S3 - Normal Distribution and confidence intervals