PHAS0051: Data Analysis Problem

PHAS0051: Data Analysis Problem Sheet 2024/25

Page 1PHAS0051 Data Analysis Problem Sheet 2024/25Four questions totalling 42 marksSubmission deadline 5pm, Monday 21st October.SoRA submission deadline 5pm, Wednesday 30th October.Submission via Turnitin assignment links on PHAS0051 Moodle page->Assessment tab.

Please make sure you convert all files to pdf before submitting and check thatyour file is readable.You may use spread sheets or programs in the calculations, but you should explain yourworkings briefly (including equations used). If you are using Python then add clear text

comments to your answer and submit a pdf not an executable program. An Excelspreadsheet is a convenient method for χ 2 calculations (use Solver for minimising).A useful resource for answering the problem sheet is the book “Measurements and their

Uncertainties” by Hughes and Hase, (See Chapter 8 for hypothesis testing and degrees offreedom). An e-book link is available on the PHAS0051 Moodle overview pagePlease make sure you read each question fully including any footnotes

See the UCL Academic Manual for late submission penalties.The use of AI is prohibited.PHAS0051: Data Analysis Problem Sheet 2024/25 NN1/10/2024Page 2

1. Mean, Variance, Standard Deviation, Standard Error on the Mean and Confidence Limits[10 Marks]UK climate records are held at the Met Office and are available at the following website:http://www.metoffice.gov.uk/public/weather/climate-historic/#?tab=climateHistoricData has been taken from this website and is presented in appendix I. The data refers to tmin(C), tmax(C), amount of rain (mm) and sunlight (hours) for the month of August, dating from1911-2023 for the station Bradford.

1. From this data set calculate separately the mean, variance, standard deviation andstandard error on the mean for both tmaxand tminover the period 1911-2022. [3]Estimate the 60% and 90% confidence limits for tmaxand tminfor the period 1911-Assume the distributions are Gaussian.[3]

1. By considering tmax

and tminfor the periods 1911-1922 and 1995-2022, discusswhether there is evidence of global warming in Bradford.[4]

2. Linear fit and 𝛘 𝟐

[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.V/volts 0.20 0.40 0.60 0.80 1.00 1.201.401.601.802.002.20I/mA

4.10 5.15 6.11 7.31 8.53 9.90 11.22 12.69 14.21 15.77 17.60The current measurements are estimated to have the same uncertainty (𝑑𝐼 = ±0.20𝑚𝐴),whilst the potential drop (V) is estimated to be without significant uncertainty.

1. Using an appropriate program, least squares fit the data to a straight line.[3]

1. Calculate the 𝜒 2 probability for the fit.[3]

1. On the same graph, plot the data with uncertainty bars and the best fit line andcomment on the outcome of the experiment in light of the 𝜒 2 probability calculatedin part ii and the graphical comparison of the best fit line and the data.

[4]PHAS0051: Data Analysis Problem Sheet 2024/25 NN1/10/2024Page 3Poisson Distribution and 𝛘 𝟐

[12 Marks] The following are results from an experiment in代 写PHAS0051: Data Analysis Problem  which a Geiger counter is used to record thenumber of particles emitted per second by a weak radioactive source. In496 repeatedobservations for a 1 second interval, counts are recorded with the following frequency:Counts 0

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

1. Using the values for the mean found in (i) calculate the theoretically expected Poissonfrequencies for each number of counts.[3]

1. Calculate the χ 2 probabilities P(χ 2 ) that the frequency distributions follow theexpected Poisson distributions with the means you have determined. You canapproximate the error on each frequency N, by a Gaussian with standard deviation√N (See Note 2 & 3 at the end of this question).[4]

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

The Poisson distribution is given by:P(r) =μ r e −μr!Where, r is the frequency and μ is the mean number of counts.

Note 1: 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 2: the probability 𝐏(𝛘 𝟐 ) is such that 𝟏 − 𝐏(𝛘 𝟐 ) is the probability that the value of 𝛘 𝟐is as low, or lower than the calculated value of, 𝛘 𝟐 . So a value of 𝐏(𝛘 𝟐 ) very close to unity means an improbably good fit – perhaps too many parameters have been used in the fit, or

the uncertainties are underestimated.PHAS0051: Data Analysis Problem Sheet 2024/25 NN1/10/2024

Page 4Best fit by Minimising 𝝌 𝟐

[10 Marks]

An experiment is carried out where the voltage, VC,across the capacitance in a resonantseries LCR circuit is measured as the frequency f, (ω is the angular frequency) is variedthrough resonance. The following estimates of the voltage versus frequency are obtained.

449.97The uncertainty in the voltage is +/- 0.15V, the frequencies are without significantuncertainty. According to AC theory the voltage is predicted to vary as:𝑉𝑐 =𝐸0𝐶(𝜔2𝑅2 + 𝐿 2(𝜔2 − 𝜔0 2 ) 2)12Where E0

 the amplitude of the AC voltage applied across the circuit which is measured to 12 volts and L and C are the inductance and capacitance in the circuit having the values30mH and 950pf. The inductance and capacitance are high precision components and theuncertainties on their values are negligible as is the value of the uncertainty on E0. ω0is thenatural frequency of the freely oscillating circuit with zero damping such that;𝑅 = 0, 𝜔0 =1√𝐿𝐶R in the circuit is only nominally known sinceit is made up of a resistance box set to 800Ωand an additional smaller contribution due to hysteresis losses in the coreof the inductor.

1. By minimising the 𝜒 2 fit of the data to the theoretical form, estimate the bestvalue for the total resistance in the circuit and the additional resistance due tohysteresis[5]

1. What is the 𝜒 2 probability for the best fit and does this value support the AC