1. Mean, Variance, Standard Deviation, Standard Error on the Mean and Confidence Limits [8 Marks]

A student wishes to measure the density ρ, of an unknown metal by measuring both the weight and volume increase when immersing a sample in a graded water container on scales. The following values were measured for the mass m, and volume V.

i.      Calculate the individual values of ρ and then the mean values for m, V and ρ from the values in the table.  [2]

iii.      Calculate the mean value of ρ from the mean values of m and V.  [1]

iv.      Calculate the  uncertainty  on the  mean value found  in  (iii)  by  propagation  of the uncertainties on the mean values of m and V.  [1]

v.      Calculate the 95% confidence level limits on ρ using the uncertainty on the mean estimated in (iv.) and assuming that the measurements are Gaussian.  [1]

2. Linear fit and X2            [10 Marks]

An experiment is performed to test whether the current through a semiconductor device I, depends linearly on the potential drop across it, V. The following data are obtained.

The current measurements are estimated to have the same uncertainty (dI = ±0.25mA), whilst the potential drop (V) is estimated to be without significant uncertainty.

i.      Using an appropriate program, least squares fit the data to a straight line. Make sure you fully label the graph. [3]

ii.      Calculate the X2  probability for the fit.  [3]

iii.      On the same graph, plot the data with uncertainty bars and the best fit line and        comment on the outcome of the experiment in light of the X 2  probability calculated in part ii and the graphical comparison of the best fit line and the data.  [4]

3. Poisson Distribution and X2      [12 Marks]

The following are results from an experiment in which a Geiger counter is used to record the number of particles emitted per second by a weak radioactive source. In 481 repeated           observations for a 1 second interval, counts are recorded with the following frequency:

i.      Calculate the mean number of counts per second (μ), the variance and the standard deviation.  [3]

ii.      Using the value for the mean found in (I) calculate the theoretically expected Poisson frequencies for each number of counts.  [3]

iii.      Calculate the χ2  probability P(χ2) that the frequency distribution follows the          expected Poisson distribution with the mean you have determined. You can            approximate the error on each frequency N, by a Gaussian with standard deviation √N (See Note 2 & 3 at the end of this question).  [4]

iv.      Plot the experimental data and theoretical Poisson distributions.  [2]

The Poisson distribution is given by:

P(r) =

Here r is the measured number of counts/s- 1 and μ is the mean number of counts/s-1 .

Note 2: the approximation that the distribution of each Poisson frequency is a Gaussian with standard deviation √N is a poor one for small N (less than 5 - see Hughes and Hase, p111      Ch8.6). In order to make the approximation reasonable when comparing the above                 distribution with theory, the contents of adjacent bins may be summed to give sufficiently    large frequencies.

Note 3: the probability P(χ2) is such that 1 − P(χ2) is the probability that the value of χ2  is as low, or lower than the calculated value of χ2 . So a value of P(χ2) very close to unity         means an improbably good fit – perhaps too many parameters have been used in the fit, or the uncertainties are underestimated.

4. Best fit by Minimising X2        [10 Marks]