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100BHW

# 100BHW - times Let p be the probability that we observe the...

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STAT 100B HWI Due Friday 5pm Problem 1 Let X 1 , ..., X n Bernoulli( p ) independently. Let S = n i =1 X i . Suppose we have two estimators of p . (1) ˆ p = S/n , and (2) ˆ p = ( S + n/ 2) / ( n + n ). (1) Calculate the bias and variance of each estimator. (2) Calculate the mean squared error of each estimator. Plot the mean squared errors of the two estimators together over the true value of p [0 , 1], for n = 5, n = 10, and n = 100 respectively. Problem 2 Suppose we roll a die 1000 times independently, and we observe the number six 180
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Unformatted text preview: times. Let p be the probability that we observe the number six each time. (1) Find the 95% conﬁdence interval of p . What is the margin of error? Find the 90% conﬁdence interval of p . (2) Test the hypothesis H : p = 1 / 6 versus the hypothesis H 1 : p > 1 / 6. Calculate the p-value. Is there evidence that the die is not fair? 1...
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