PHYS5032 Techniques for Sustainability Analysis
Total points: 10
Homework Sheet 1: Random variables
1. (A) Design a normally distributed random variable that can generally reflect the variation of global average temperature during 1851 to 1971 in Excel with N=5 values.
• Note that Excel’s RAND function does not generate a normally distributed dataset and you would need to google how to do this).
• You will also need to define the initial expected value and the initial variance to generate your data set.
- Tips: you can calculate the two terms based on records of global average surface temperature which are usually presented as anomalies1 rather than as absolute temperatures. Such historical global average surface temperature data are provided on the Canvas Homework page.
- Once finished, calculate the expected value and the variance or your actual data set. Are they the same as the ones that you used for defining the random number generator? If not, why not? (2 points)
2. (A) Use the random variable of question No. 1 and augment the variable by another 35 values. These should have approximately (but not identically) the same distribution as the first 5. Calculate the expected value and the variance for this new experiment with N=40. What changes? (1 points)
3. (A) What is the confidence interval for 95% for the N=40 experiment? (1 point)
4. (A) Imagine the hypothetical case that after you carried out another 200 events (your sample set is now at N=240), the expected value of the random variable stays the same as the N=40 experiment, but the variance is half of the N=5 experiment.
Questions:
4.1 What would happen to the confidence interval for 95% comparing to the N=40 experiment (1 point)
4.2 Please provide a factor how much larger or smaller the interval would be (make sure including you calculation process/equations)? (2 points)
5. In your own words (A): Would the expected value and variance of the global average temperature change from 1971 to 2023, compared to the ones you defined in Question 1? If yes, by how much and why? (3 points, within 400 words).
Note: Some of the tasks in this (and other) homework sheets require that you use computer programs to perform calculations of visualize your data. Excel, OpenOffice, R, or Matlab are excellent programs to use. Each homework sheet has a number of questions that are marked with (A). These questions will be part of your assignment.