L.C. Higher, Paper 2, 2011 – Pilot Q7 (75 marks) Solutions Usual format. How many have not yet covered P&S? Not much theory to be covered. Will introduce it and then you just need a clear head to answer the questions. 7 questions in handout. If you find the pace slow you can work ahead, but more important to use the tutorial to consolidate your understanding than just work on your own. If you get to the end of the 7 questions ask a tutor for a copy of an additional question. All slides will be on the internet, including solution to additional question. How many don’t have access to the internet? Everybody should have a copy of formulae and tables. Always use it so that finding what you need in it becomes second nature. It only costs €4 from schoolbooks.ie. You will certainly need it next week for Statistical Inference. Safety precautions to be covered? The fire exits are…….

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) Create a suitable graphical representation of the distribution. How? Draw a histogram A histogram is a graph that shows the frequency, or the number of times, something happens within a specific interval. And we want to show the frequency of earthquakes in given time periods.

Time in days from previous earthquake L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) First adjust the table given so that the time periods are all of the same length. i.e. the table becomes: Watch out for data where the groupings (e.g. groups within time periods) are not consistent and need to be rearranged. Time in days from previous earthquake 0 - 100 100 -200 200 - 300 300 - 400 400 - 500 500 - 600 600 - 700 700 - 800 800 - 900 900 - 1000 1000 - 1100 1100 - 1200 1200 - 1300 Number of earthquakes 31 24 12 14 8 7 5 6 2·5 1 We were told in the period 800 – 1000 days from previous, 5 earthquakes occurred. So we assume half happened in 800 – 900 days from previous earthquake, and the other half happened in 900 – 1000. Likewise, split the 3 earthquakes between 1000 - 1300 days into three equal parts, 1 earthquake in each of the hundred days.

Time in days from previous earthquake L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) Time in days from previous earthquake 0 - 100 100 -200 200 - 300 300 - 400 400 - 500 500 - 600 600 - 700 700 - 800 800 - 900 900 - 1000 1000 - 1100 1100 - 1200 1200 - 1300 Number of earthquakes 31 24 12 14 8 7 5 6 2·5 1 Our histogram from the above data is:

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) (ii) The distribution is skewed to the right. The median is the value where half the values are less and half of the values are greater. This would correspond to the value where half of the area under the histogram is to the left and half of the area under the histogram is to the right. So, visually, you can estimate the median by finding the point where the area to the left and right are about equal. So we have the area representing 115 earthquakes, at which point are 57.5 earthquakes (115 ÷ 2 = 57.5) on one side of the histogram? - > At roughly 220 days.

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) (iii) This is not a normal distribution since it is skewed to the right and small values are common but some large ones can occur. The normal distribution is a symmetric, bell shaped distribution with no bias (or skew) to the left or to the right, looking like this: Continuous probability distribution. Area under curve between two points give probability of a value between those points. Will talk more about the normal distribution next week.

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) (iv) We want the best estimate of the probability that an earthquake occurs between 100 – 200 days since the last one. i.e. given our data, from all the occurrences of earthquakes, how many times did it occur in the 100 – 200 band? answer: 24 times out of 115 total earthquakes i.e. 24/115 ≈ 0.2

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) (v) [This is really quite a general question – with marks given where any suitable option is put forward] • They could have looked at the number of earthquakes each year, or some other interval of time (e.g. distribution of earthquakes per decade, per year, etc.) • They could have redefined serious earthquakes as earthquakes greater than a certain magnitude; earthquakes in less populated areas are not included. • The data set could have been broadened to include less serious earthquakes. This could result in a different pattern.

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) (b) (i) The statement is too deterministic – strong earthquakes don’t always cause tsunamis and weak ones sometimes do. A better statement would be “Strong earthquakes are more likely to cause tsunamis than weaker ones.” (ii) Reading from the graph, about 103 of these did and about 156 didn’t. So the probability is the number that did over the total that either did or didn’t (there are only 2 options). So probability is 103/259 ≈ 0·4

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) (iii) Consider the next six earthquakes of magnitude at least 7·5. Find an estimate for the probability that at least four of them will cause a tsunami, assuming that these six events are independent of each other.

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) (iii) Consider the next six earthquakes of magnitude at least 7·5. Find an estimate for the probability that at least four of them will cause a tsunami, assuming that these six events are independent of each other. Answer Again, just use the graph to check the numbers in the range above 7.5. So we have: Tsunami : 142 + 60 + 8 =210 No tsunami: 139 + 36 + 7 = 182 Total : 210 + 182 = 392 Our probability is the total tsunamis over the overall total: p ≈ 210/392 ≈ 0·54

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks) But we aren’t finished yet. The question asks us to check when at least four of six earthquakes cause a tsunami. This could be 4 of 6 earthquakes causing tsunamis, 5 of 6 earthquakes causing tsunamis or all 6 of 6 earthquakes causing tsunamis. This is:

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks)

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks)

L.C. Higher, Paper 2, 2011 – PM Pilot Q7 (75 marks)