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Assume that we have a finite population of 5 elements on which we measured a random...
7.109 Sample variance: Let X be a random variable with finite variance. Supposem don't know the...
7.109 Sample variance: Let X be a random variable with finite variance. Supposem don't know the variance of X and want to estimate it. You take a random sample, A1, sample, X1,..., X from the distribution of X and set S = (n − 1)-'L'=(X; - X)2. Show that the random variable 2-which is called the sample variance based on a sample of size n-is an unbiased estimator of oz.
Suppose that a random sample of size 1 is to be taken from a finite population...
Suppose that a random sample of size 1 is to be taken from a finite population of size N. Answer parts (a) through (c) below. a. How many possible samples are there? A. N + 1 B. N-1 0 C. N D. 1 b. Identify the relationship between the possible sample means and the possible observations of the variable under consideration. Choose the correct answer below. A. Each possible sample mean is equal to an observation, only if the population...
For the population of N = 5 units of Exercise 3 of Chapter 2 (a) Compute...
For the population of N = 5 units of Exercise 3 of Chapter 2
(a) Compute directly the variance var (y) of the sample mean and
the variance var( m ) of the sample median.
(b) From each sample, compute the sample variance s 2 and the
estimate var (y) of the variance of the sample mean. Show that the
sample variance s 2 is unbiased for the √ finite-population
variance σ 2 but that the sample standard deviation 2...
Consider a finite population with five elements labeled A, B, C, D, and E. Five possible...
Consider a finite population with five elements labeled A, B, C, D, and E. Five possible simple random samples of size 4 can be selected. a. Choose the correct list of the 5 samples. - Select your answer - b. Using simple random sampling, what is the probability that each sample of size 4 is selected? C. Suppose we use Excel's RAND function to assign random numbers to the five elements: A(0.2826) B(0.9735) C(0.527) D(0.3673) E(0.5364). List the simple random...
I. Consider a random sample of size n from a normal population with variance σ (n-1s2...
I. Consider a random sample of size n from a normal population with variance σ (n-1s2 (a) Find and identify the distribution of S2. Hint: what do you know about (1) (b) Find EIS] and Var(S2)
Probem.ume that the distribution 4o, Lat bethe sample mean of a random sample of n- 16...
Probem.ume that the distribution 4o, Lat bethe sample mean of a random sample of n- 16 observations of X a) True or false: (1) E(X) E(X)? (2) E(X) = X? Briefly explain your reasoning. b) Give the values of E(X) and Var(X) c) Find P(44.42 X48.98). d) Let s2 be the sample variance of the 16 observations. Find constants a and b so that P(a $2-b) = 0.9. Hint: What is the distribution of s2? (note that it is not...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y)...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
Consider a finite population with five elements labeled A, B. C, D, and E. Ten possible...
Consider a finite population with five elements labeled A, B. C, D, and E. Ten possible simple random sampies of size 2 can be Selected. a. List the 10 samples beginning with A, B, C, D, and E. Using simple random sampling, what is the probability that each sample of size 2 is selected? b. Assume the number 1 corresponds to A, the number 2 corresponds to B, and so on. Which of the following represents the simple random sample...
Where p denote the population mean of the original random variable 5.7 Problems . Assume X is a n...
where p denote the population mean of the original random variable 5.7 Problems . Assume X is a normally distributed random variable with mean u and stan- dard deviation σ. A sample of size n-5 from this distribution is given as 1. Assume we are interested in the properties of the mean of the sam- pling distribution of the sample mean. Describe why this quantity is a 2. State an estimator for the parameter given in question 1. Use this...
Suppose that X1, X2n is a random sample of size 2n from a population with mean μ and variance σ2 ...
Suppose that X1, X2n is a random sample of size 2n from a population with mean μ and variance σ2 for which the first four moments are finite. Find the limiting distribution to which the following random sequence converges in probability: 7l
Suppose that X1, X2n is a random sample of size 2n from a population with mean μ and variance σ2 for which the first four moments are finite. Find the limiting distribution to which the following random sequence...
7.109 Sample variance: Let X be a random variable with finite variance. Supposem don't know the variance of X and want to estimate it. You take a random sample, A1, sample, X1,..., X from the distribution of X and set S = (n − 1)-'L'=(X; - X)2. Show that the random variable 2-which is called the sample variance based on a sample of size n-is an unbiased estimator of oz.
Suppose that a random sample of size 1 is to be taken from a finite population of size N. Answer parts (a) through (c) below. a. How many possible samples are there? A. N + 1 B. N-1 0 C. N D. 1 b. Identify the relationship between the possible sample means and the possible observations of the variable under consideration. Choose the correct answer below. A. Each possible sample mean is equal to an observation, only if the population...
For the population of N = 5 units of Exercise 3 of Chapter 2
(a) Compute directly the variance var (y) of the sample mean and
the variance var( m ) of the sample median.
(b) From each sample, compute the sample variance s 2 and the
estimate var (y) of the variance of the sample mean. Show that the
sample variance s 2 is unbiased for the √ finite-population
variance σ 2 but that the sample standard deviation 2...
Consider a finite population with five elements labeled A, B, C, D, and E. Five possible simple random samples of size 4 can be selected. a. Choose the correct list of the 5 samples. - Select your answer - b. Using simple random sampling, what is the probability that each sample of size 4 is selected? C. Suppose we use Excel's RAND function to assign random numbers to the five elements: A(0.2826) B(0.9735) C(0.527) D(0.3673) E(0.5364). List the simple random...
I. Consider a random sample of size n from a normal population with variance σ (n-1s2 (a) Find and identify the distribution of S2. Hint: what do you know about (1) (b) Find EIS] and Var(S2)
Probem.ume that the distribution 4o, Lat bethe sample mean of a random sample of n- 16 observations of X a) True or false: (1) E(X) E(X)? (2) E(X) = X? Briefly explain your reasoning. b) Give the values of E(X) and Var(X) c) Find P(44.42 X48.98). d) Let s2 be the sample variance of the 16 observations. Find constants a and b so that P(a $2-b) = 0.9. Hint: What is the distribution of s2? (note that it is not...
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
Consider a finite population with five elements labeled A, B. C, D, and E. Ten possible simple random sampies of size 2 can be Selected. a. List the 10 samples beginning with A, B, C, D, and E. Using simple random sampling, what is the probability that each sample of size 2 is selected? b. Assume the number 1 corresponds to A, the number 2 corresponds to B, and so on. Which of the following represents the simple random sample...
where p denote the population mean of the original random variable 5.7 Problems . Assume X is a normally distributed random variable with mean u and stan- dard deviation σ. A sample of size n-5 from this distribution is given as 1. Assume we are interested in the properties of the mean of the sam- pling distribution of the sample mean. Describe why this quantity is a 2. State an estimator for the parameter given in question 1. Use this...
Suppose that X1, X2n is a random sample of size 2n from a population with mean μ and variance σ2 for which the first four moments are finite. Find the limiting distribution to which the following random sequence converges in probability: 7l
Suppose that X1, X2n is a random sample of size 2n from a population with mean μ and variance σ2 for which the first four moments are finite. Find the limiting distribution to which the following random sequence...
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