Please show your work with each question. # Write the whole process of answering the question. 1. Your friend has custom-made a rigged six-sided die. Instead of each side coming up with probability not the faces of the die have the following probabilities: -___ 1 i 3 l E 10 2t: 5 1 .FLIIl Ln 4:. * Note: The above probabilities are different from probabilities of the normal die. Calculate the following probabilities by hand. and show your work. (a) What's the probability of rolling a number that is even and is greater than 3'? (b) What's the probability ofrolling a number that is even or is greater than 3? (e) Rolling the die once. 1What's the expected value? 1What's the variance? 2. Use the table of Z scores (you can nd the Z-table in Rutgers Canvas of this class) to answer the following questions. [a] SAT math scores are normally distributed with mean 520 and standard deviation 100. 'vVhat 'percentage' of test takers score at least a T00? {13} Height of NBA players is normally distributed' with a mean of 78 inches and a standard deviation of4 inches. You want to find the 'shortest' 2% of NBA players: what would be the cutoff point? {c} Daily worldwide oil production is normally distributed. with a mean of 66 million barrels and a standard deviation: of 6 million. What's the probability that production today will be 'between' .54 million barrels and 63 million barrels? 3. Suppose you knew that atrtct't 45% of the American population approved of the president's job performance. In probability terms. this means that for a random American. the president's approval is distributed Bernoulli (Y = l for approve. vs. Y = U for disapprove] with it = 0.45. (a) What are the expected value and variance of the president approval in the population? [b] Now suppose a polling firm is hired to randomly sample Americans. asking them whether they approve of the president [Y = l} or not [Y = 0}. The proportion of respondents in the sample who approve of the president is simply calculated as the average on over these observations. (it If the polling firm calculates the proportion based on a sample of 100 Americans. what does the Central Limit Theorem tell us about the sampling distribution for the proportion approving of the president? In Particular. what are the means and standard error of the sample proportion statistic? (*Think about Population mean and Sample mean as well as Population variance and Sample Varianco. According to CLT. they would be same or not) (ii) Repeat the analysis (i). but for a sample of SDI). 4. Construct confidence intervals (two-aided) from the following sample data, for the given condence level. (a) 95% confidence interval for a mean: Known s.d.: 0': 2 Sample mean: X = 5 Sample size = n = 450 {b} 80%- confidence interval for a mean: Sample e.d.: s = 30 Sample mean: X = 420 Sample size: n = 20 t c] 99% confidence interval for a proportion: Sample proportion: p = 0.33 Sample size: n = 890 {d} 90% confidence interval for a proportion: Sample proportion: p = 0.63 Sample size: n = 63