Question: A altimetric satellite (Jason-1) measures the height of the ocean's surface and is calibrated by repeated passes over a point on the Earth where precise, ground (here ocean) based sensors provide an independent measure of the sea surface height. Each pass over this reference location gives another measurement of the altimeter bias, which is then used to compute an average altimeter bias. If we assume that our measurement errors are normally distributed with a mean of zero, then the uncertainty of the true mean of the bias, \(\sigma_{b}\), is

\({{\sigma }_{b}}=\frac{{{z}_{\alpha /2}}{{s}_{b}}}{\sqrt{n}}\)

where \(z\) is the standard normal deviate, \(s_{b}\) is the estimated standard deviation of the measurements and \(\mathrm{n}\) is the number of measurements (i.e. the number of calibration passes)

1. If we are required to know the true mean of the altimeter bias to within \(\pm 2 \mathrm{~cm}\) and we estimate the uncertainty of the individual measurements to be \(3 \mathrm{~cm}\), compute the number of measurements required for \(90 \%, 95 \%\) and 99% confidence levels (round off the next largest integer).
2. What number of measurements is required for the \(90 \%, 95 \%\) and \(99 \%\) confidence levels if we want to know the true mean to within \(1.5 \mathrm{~cm} ?\)
3. If the satellite is in a ten repeat orbit (hence the calibration measurement is possible once every 10 days), and we are allowed 240 days for collecting calibration data. If we assume that \(50 \%\) of the calibration measurements are successful, then what confidence limits are achievable for each of the cases examined in a and \(b\).

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