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3. A university wants to know what proportion of students are regular bike riders so that they can install an appropriate number of bike racks. The spreadsheet bike. csv contains the result of survey conducted in 20 classes. Here N is a class size and n is a number of regular bike riders in the class. The below code reads the spreadsheet. bike=read.csv(‘bike.csv’) Let θ be a probability that a student is a regular bike rider. The university thinks that a Beta distribution Beta(30,10) is a sensible distribution for θ based on their previous study. (a) Find a posterior distribution for θ given the survey result. [2] (b) Find the posterior mean and the 0.9 credible interval for θ. [2]

(c) Describe how the degrees of belief about θ is updated from the prior belief by information in the survey result. [2] (d) One more survey is conducted for the new class of 73. Given the existing survey result, find the expected value and the variance for a number of regular bike riders in the new class. [3]

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