Question: This altimeter height bias is just one of many errors that contribute to the overall error budget of the altimetric height. Let's say that we wish to know if the height error is less than \(13 \mathrm{~cm}\). Set up the null hypothesis \(\left(\mathrm{H}_{0}\right)\) that the true mean is less than \(13 \mathrm{~cm}\). This means that we do not wish to erroneously reject the hypothesis more than \(\alpha \times 100 \%\) of the time (where \(1-\alpha\) is the confidence interval). If we define our hypothesis limit, HL, as

then if the mean of our measurements is greater than \(\mathrm{HL}\), we should reject \(\mathrm{H}_{0}\) and conclude that the mean is greater than \(13 \mathrm{~cm}\) with a probability \(\alpha\) of being wrong. If after 9 measurements are taken each measurement is assumed to have an uncertainty of \(3 \mathrm{~cm}\) and the mean height error is estimated to be \(15 \mathrm{~cm}\), do we accept or reject the hypothesis for \(\alpha=0.10 ?\) For \(\alpha=0.05\) ? For \(\alpha=0.01\) ?

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