Find the 95% confidence interval for the true fraction p. Exercise 4.8. In a million rolls of a biased die the number 6 shows up 180,000 times.

172 Approximations of the binomial distribution

Hint. You will need the continuity correction for this.
(ETcercise 4.4, Liz is standing on the real number line at position 0. She rolls a die repeatedly. If the roll is 1 or 2, she takes one step to the right. If the roll is 3, 4, 5 or 6, she takes two steps to the right. Let Xn be Liz’s position after n rolls of the die. Estimate the probability that X90 is at least 160.
Section 4.2
xercise 4. Consider the setup of Exercise 4.4. Find the limits below and explain your answer.
(a) Find lim 11X, > 1.6n). n—> oo (b) Find lim P(X,, > 1.7n).
Section 4.3
oo
Exercise . A pollster would like to estimate the fraction p of people in a population who intend to vote for a particular candidate. How large must a random sample be in order to be at least 95% certain that the fraction p of positive answers in the sample is within 0.02 of the true p? Exercise. A political interest group wants to determine what fraction p E (0, 1) of the population intends to vote for candidate A in the next election. 1000 randomly chosen individuals are polled. 457 of these indicate that they intend to vote for candidate A. Find the 95% confidence interval for the true fraction p. Exercise 4.8. In a million rolls of a biased die the number 6 shows up 180,000 times.

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