Question 3 Consider the MacKay coin-flipping example from lectures. In this question you will do hypothesis testing the 'second way' using marginal likelihoods. A coin was flipped N = 250 times and heads was the result x = 140 times (with tails appearing 110 times). We want to know whether the coin flipping was fair or not. Let 0, the unknown parameter be the probability of heads on each flip, and let Ho be the hypothesis that = 0.5, and H₁ the hypothesis that 00.5. Use R for this question. 1 (a) Create a vector of values for ranging from 0 to 1 in steps of 0.001. Also, create a prior for given H₁ that is proportional to dnorm (theta, mean=0.5, sd=0.1). This models the idea that we only would have expected a slightly biased coin tossing procedure. (b) Calculate the marginal likelihood P(x = 140 | H₁). (c) Calculate a 95% credible interval for under the assumption of H₁. (d) Calculate the likelihood for Ho and therefore the Bayes Factor for H₁ over Ho. (e) Find the posterior probability of Ho if it has a prior probability of 0.5.
Question 3 Consider the MacKay coin-flipping example from lectures. In this question you will do hypothesis testing the 'second way' using marginal likelihoods. A coin was flipped N = 250 times and heads was the result x = 140 times (with tails appearing 110 times). We want to know whether the coin flipping was fair or not. Let 0, the unknown parameter be the probability of heads on each flip, and let Ho be the hypothesis that = 0.5, and H₁ the hypothesis that 00.5. Use R for this question. 1 (a) Create a vector of values for ranging from 0 to 1 in steps of 0.001. Also, create a prior for given H₁ that is proportional to dnorm (theta, mean=0.5, sd=0.1). This models the idea that we only would have expected a slightly biased coin tossing procedure. (b) Calculate the marginal likelihood P(x = 140 | H₁). (c) Calculate a 95% credible interval for under the assumption of H₁. (d) Calculate the likelihood for Ho and therefore the Bayes Factor for H₁ over Ho. (e) Find the posterior probability of Ho if it has a prior probability of 0.5.
Chapter8: Sequences, Series,and Probability
Section8.7: Probability
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Transcribed Image Text:Question 3
Consider the MacKay coin-flipping example from lectures. In this question you will do hypothesis
testing the 'second way' using marginal likelihoods. A coin was flipped N = 250 times and
heads was the result x = 140 times (with tails appearing 110 times). We want to know whether
the coin flipping was fair or not. Let 0, the unknown parameter be the probability of heads on
each flip, and let Ho be the hypothesis that = 0.5, and H₁ the hypothesis that 00.5. Use R
for this question.
1
(a) Create a vector of values for ranging from 0 to 1 in steps of 0.001. Also, create a prior
for given H₁ that is proportional to dnorm (theta, mean=0.5, sd=0.1). This models
the idea that we only would have expected a slightly biased coin tossing procedure.
(b) Calculate the marginal likelihood P(x = 140 | H₁).
(c) Calculate a 95% credible interval for under the assumption of H₁.
(d) Calculate the likelihood for Ho and therefore the Bayes Factor for H₁ over Ho.
(e) Find the posterior probability of Ho if it has a prior probability of 0.5.
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