Question:

1. Suppose 40% of people carry the gene for a particular disease. Further suppose that you take a random sample of n = 120 people, and let be the proportion of people in your sample who carry the gene. Show your work below.

a. What are the mean and standard deviation of the sample proportion ? Show your work

Mean:

Standard deviation:

b. What is the approximate probability that exceeds 0.46?

2. "For the first time in Gallup's 34-year trend, a majority of Americans say that life imprisonment with no possibility of parole is a better punishment for murder than the death penalty is." The article says that "60% say life imprisonment the better punishment." The article says that it is based on a random sample of 1,506 U.S. adults and that the study has a margin of error of 3 percentage points for 95% confidence in its conclusions. What is the 95% confidence interval for the percentage of all U.S. adults who believe that life imprisonment with no possibility of parole is a better punishment for murder than the death penalty is? You do not need to interpret it. Show your work

3. In the article AA Multifactorial Intervention Program Reduces the Duration of Delirium, Length of Hospitalization, and Mortality in Delirious Patients@ (Journal of the American Geriatrics Society, Vol. 53, No. 4, pp. 622-628). M. Lundstrom et al.) investigated whether education programs for nurses improve the outcomes for their older patients. Assume that the mean length of stay on the intervention ward is 9.3 days and the standard deviation is 8.2 days. Suppose you take a sample of 80 patients on the intervention ward. Show your work on all parts below.

a. What are the mean and standard deviation of the sample mean ? Show your work

Mean:

Standard deviation:

b. What is the probability that the mean length of stay of the 80 patients is less than 7 days?

4. Suppose that we wish to estimate the proportion of Americans who approve of the job the President is doing. Assume we know that the proportion should be near the most recent approval rating of 43%, or 0.43. Let's say that we want to estimate p to within 0.04 with 95% confidence. What sample size is required? Show your work. Show your work

Question 5 Let A = ] 1 (a) Calculate A A and find its spectral decomposition. (b) Calculate AA and find its spectral decomposition. (c) Compare the nonzero eigenvalues of A A and those of AAT.Find the spectral decomposition of es . A = 0 0 1+ Write each orthogonal projection matrix as a single matrix.9.14 The beta function is B(r, s) = I()I(s) T(r+s) , where I' is the gamma function. Equivalently B(r, s) = tr - 1 ( 1 - t) s - 1dt ( r > 0, s > 0 ). X is said to have a beta distribution if the density f of its distribution measure is or-1(1 - x) $-1 f (20) = if 0 1. Show that for X having a beta distribution with parameters (r, s) (r > 0, s > 0), then B(r + k, s)_ [(r+ k) I(rts) B(r, s) [(r) I (r t s + k)' for k > 0. Deduce that r E(X} = rts rs 62 ( X ) = (rt s ) ? (rt s+ 1 ) The beta distribution is a rich family of distributions on the interval [0, 1]. It is often used to model random proportions.1. The random variable X has probability density function caVI-x, if0