please please help me out for these question below. They are very important to me. As I am not good on Statistic and I don't have the model answers for them, I really want someone who can show me the answers about them. Therefore, I can check if i was doing the questions in the right way.

Don't have to do all of them, just please do the one you know. Thank you very much!!!

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Question 1 -- Let X, Y and Z be random variables.

(a) Define the covariance Cov(X, Y ).

(b) Prove that Cov(Z,X + Y ) = Cov(Z,X) + Cov(Z, Y ).

(c) Suppose that X and Y are independent, with Var(X) = 25 and Var(Y ) = 6,

and that Z = X + 2Y . Find the correlation between X and Z.

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Question 2---Suppose that we have data x1, . . . , xn.

(a) Name two measures of the location of the data, and explain how they are calculated.

Explain which of these is more appropriate if the data consist of the annual

income of n people.

(b) Name two measures of the spread of the data, and explain how they are calculated.

(c) For i = 1 . . . , n, let yi = axi + b, where a and b are constants and a is not equal to 0. Let (mean of x)¯x and (standard deviation of x) sx be the sample mean and sample standard deviation, respectively, of x1, . . . , xn, and let ¯y and sy be the sample mean and sample standard deviation, respectively, of y1, . . . , yn. Find ¯y and sy in terms of ¯x and sx.

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Question 3---Let X and Y be random variables all of whose values are non-negative integers.

(a) Define the probability generating function of X.

(b) State a theorem linking the probability generating functions of X, Y and X+Y in the case that X and Y are independent of each other.

(c) Suppose that X Poisson(入). Find the probability generating function of X.

(d) Suppose that X and Y are independent of each other, X ~Poisson(入) and

Y~Poisson(μ). Find the distribution of X + Y .

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Question 4---A Minitab worksheet contains the following information

about students enrolled for MTH4106 Introduction to Statistics. Column C1 contains the number of Minitab practical sessions that they have attended. Column C2 contains their marks on the first test. Column C3 contains their gender, coded as 1 for male and 2 for female.

(a) State what type of variable is in each of these columns.

(b) Explain, using a sketch, how to plot these three variables on a single diagram.

(c) Explain briefly how this can be done in Minitab.

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Question 5---In a large population of N adults, let m be the number who are obese. The number N is known, but the number m is not known. A random sample of n people is chosen, where n is much smaller than N. Let X be the number of obese people in the sample.

(a) State the distribution of the random variable X. Hence write down the expectation and variance of X.

(b) Let Y = NX/n. Show that Y is an unbiased estimator of m.

(c) Find the standard deviation of Y .

(d) Suppose that N = 2, 000, 000 and n = 1, 000. To what accuracy should the estimate of m be reported?

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Question 6 ---A marketing manager in a firm that makes washing-up liquid wants to find out if the lemon-scented and mint-scented versions are equally

liked by consumers. If they are not equally liked, then the firm may replace the less popular one by a new version.

Let p be the proportion of customers who prefer the lemon-scented to the mint-scented version. Out of 250 consumers who were interviewed, 145 said that they preferred the lemon-scented washing-up liquid, while the other 105 said that they preferred the mint-scented version.

(a) State appropriate null and alternative hypotheses.

(b) Carry out an appropriate hypothesis test at significance level 0.01. Report

your conclusions.

(c) What sort of method of data collection was this? Give one concern that a

statistician might have about the way the data were collected.