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| − | A 95 per cent confident for a population mean looks like.
| + | Confidence interval |
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| − | <math>\huge \mu = {\bar x}\pm1.96\frac{\sigma}{\sqrt n}</math>
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| − | <img src="/static/files/MBI/Module%207/stdnormaldist.jpg" width=500
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| − | </html>
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| − | In practice we seldom know that <math>\sigma</math> is known. Therefore most of the times an estimate of standard deviation in the sample <math>s</math> (sometimes denoted as <math>\Large\hat\sigma</math>) is used:
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| − | <math>\huge \mu = {\bar x}\pm z\frac{s}{\sqrt n}</math>
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| − | If we use:
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| − | <math>\huge e=z \frac {s}{\sqrt n}</math>
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| − | We can make a conclusion on determining the minimum sample size given a requested error level and confidence level::
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| − | <math>\huge n=(\frac {zs}{e})^2</math>
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| − | Adding up confidence intervals for the difference of means is given by:
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| − | <math>\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})}</math>
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| − | 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:
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| − | <math>\huge \sqrt {(1-\frac {n}{N})}</math>
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| − | So corrected with the finite population correction factor a 95 per cent confident for a population mean looks like:
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| − | <math>\huge \mu = {\bar x}\pm 1.96 \sqrt {(1-\frac {n}{N})} \frac{s}{\sqrt n}</math>
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| − | The [[T-distribution|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.
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| − | Part of [[M7-Quantitative Decision Methods-S3 - Normal Distribution and confidence intervals|M7-Quantitative Decision Methods-S3 - Normal Distribution and confidence intervals]]
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| | [[Category:mbi]] | | [[Category:mbi]] |
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