Question

Let X1, ..., Xn be a random sample from a Bernoulli(p) distribution. Define the sample proportion p-hat = (X1 + ... + Xn) / n. (a) According to the Central Limit Theorem, what is the approximate probability distribution of p-hat for large n? (b) Give an expression for the standardized sample proportion and argue why it is a pivotal quantity. (c) Use your result in part (b) to derive an approximate large-sample (1-Î±)Ã—100% confidence interval for the population proportion p. Hint: We cannot know the true variance of p-hat from sample data, so you will need to replace p by p-hat in the expression for the variance of p-hat. Note that this will not change the approximate distribution of the pivotal quantity. (d) A Jan. 31, 2014, Gallup article reported that "Russians See Gold in Sochi Olympic Games Yet Many Concerned About Corruption" (http://www.gallup.com/poll/167138/russians-gold-sochi-olympic-games.aspx). The article reports on results of a poll of a random sample of 2,000 adults in Russia. In the sample, 66% responded that they thought the Olympic games would increase corruption. i. Using your formula from part (c), construct an approximate 95% confidence interval for the true proportion of all adult Russians that believe the Olympic games will increase corruption. ii. Give an interpretation of your interval in context, i.e., "We are 95% confident that ." iii. Explain what the phrase "95% confident" means in your interpretation.

          Let X1, ..., Xn be a random sample from a Bernoulli(p) distribution. Define the sample proportion p-hat = (X1 + ... + Xn) / n. 

(a) According to the Central Limit Theorem, what is the approximate probability distribution of p-hat for large n?
(b) Give an expression for the standardized sample proportion and argue why it is a pivotal quantity.
(c) Use your result in part (b) to derive an approximate large-sample (1-Î±)Ã—100% confidence interval for the population proportion p. Hint: We cannot know the true variance of p-hat from sample data, so you will need to replace p by p-hat in the expression for the variance of p-hat. Note that this will not change the approximate distribution of the pivotal quantity.
(d) A Jan. 31, 2014, Gallup article reported that "Russians See Gold in Sochi Olympic Games Yet Many Concerned About Corruption" (http://www.gallup.com/poll/167138/russians-gold-sochi-olympic-games.aspx). The article reports on results of a poll of a random sample of 2,000 adults in Russia. In the sample, 66% responded that they thought the Olympic games would increase corruption.
i. Using your formula from part (c), construct an approximate 95% confidence interval for the true proportion of all adult Russians that believe the Olympic games will increase corruption.
ii. Give an interpretation of your interval in context, i.e., "We are 95% confident that ."
iii. Explain what the phrase "95% confident" means in your interpretation.

Added by William R.

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