Homework Answers
It is a problem of SRSWOR, so
the s^2 is the sample variance and S^2 is the population variance
here. We here check the unbiasedness of the two.
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1. Consider the following population of N 5 sampling units with characteristic of interest y Sampling...
Show that E(s2)-σ2 in simple random sampling, where the sample variance s* is defined with n...
Show that E(s2)-σ2 in simple random sampling, where the sample variance s* is defined with n 1 in the denominato 2 is defined with N-1 in the denominator. [Hint: Write yi-y as yi-μ- σ-μ), verify that r and the population variance σ 17 and cither take espectation over all pouible samples or define an indicator variable for each unit, indicating whether it is included in the sample.]
For purposes of studying sampling distribution, we consider a small population of N = 4 units,...
For purposes of studying sampling distribution, we consider a small population of N = 4 units, labeled 1, 2, 3, 4, with respective y-values yı = 3, y2 = 1, y3 = 0, y4 = 5. (c) Plan 2: Consider a simple random sample with replacement (SRSWR) design with sample size n=2. (i) Find the number of possible SRSs of size n = 2. List every possible sample. For each sample, what is the probability that it is the one...
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...
I1. Follow the steps below to show that the pooled estimator $p is an unbi- ased...
I1. Follow the steps below to show that the pooled estimator $p is an unbi- ased estimator for the common standard deviation of two independent sam ples Let Yi, Yi2, ..., Yini denote the random sample of size n from the first population with population mean μ| and population variance σ, and let Y21, Y22, ..., Y2na denote an independent random sample of size n2 from the second population with population mean μ2 and population mean ơ3. Sup- pose that...
CLUSTER SAMPLING WITH ESTIMATION Suppose a population of size N is divided into K- N/M groups of size M. We select a sample of size n -km the following way: » First we select k groups out of K groups...
CLUSTER SAMPLING WITH ESTIMATION Suppose a population of size N is divided into K- N/M groups of size M. We select a sample of size n -km the following way: » First we select k groups out of K groups by simple random sampling . We then select m units in each group selected on the first step by simple random sampling . The estimate of the population mean is the average Y of the sample. Let μί be the...
Assume a population of 1, 4, and 10. Assume that samples of size n = 2...
Assume a population of 1, 4, and 10. Assume that samples of size n = 2 are randomly selected with replacement from the population. Listed below are the nine different samples. Complete parts a through d below. 1,1 1,4 1,10 4,1 4,4 4,10 10,1 10,4 10,10 o a. Find the value of the population standard deviation o. (Round to three decimal places as needed.) b. Find the standard deviation of each of the nine samples, then summarize the sampling distribution...
4 and 5 samples, the other in small samples. Which is which? Explain. (d) Suppose we...
4
and 5
samples, the other in small samples. Which is which? Explain. (d) Suppose we know that the 5 values are from a symmetric distribution. Then the sample median is also unbiased and consistent for the population mean. The sample mean has lower variance. Would you prefer to use the sample 4. Suppose Yi, Y, are iid r ables with E(n)-μ, Var(K)-σ2 < oo. For large n, find the approximate 5. Suppose we observe Yi...Yn from a normal distribution...
1. Select all true statements about sample mean and sample median. A) When the population distribution...
1. Select all true statements about sample mean and sample median. A) When the population distribution is skewed, sample mean is biased but sample median is an unbiased estimator of population mean. B) When the population distribution is symmetric, both mean and sample median are unbiased estimators of population mean. C) Sampling distribution of sample mean has a smaller standard error than sample median when population distribution is normal. D) Both mean and median are unbiased estimators of population mean...
Assume a population of 2, 4, and 9. Assume that samples of size n 2 are...
Assume a population of 2, 4, and 9. Assume that samples of size n 2 are randomly selected with replacement from the population. Listed below are the nine different samples. Complete parts a through d below 4.9 9,9 2,2 2,4 2,9 4,4 9,2 9.4 a. Find the value of the population standard deviation σ (Round to three decimal places as needed.) b. Find the standard deviation of each of the nine samples, then summarize the sampling distribution of the standard...
Please give detailed steps. Thank you. 5. Let {X1, X2,..., Xn) denote a random sample of...
Please give detailed steps. Thank you.
5. Let {X1, X2,..., Xn) denote a random sample of size N from a population d escribed by a random variable X. Let's denote the population mean of X by E(X) - u and its variance by Consider the following four estimators of the population mean μ : 3 (this is an example of an average using only part of the sample the last 3 observations) (this is an example of a weighted average)...
Show that E(s2)-σ2 in simple random sampling, where the sample variance s* is defined with n 1 in the denominato 2 is defined with N-1 in the denominator. [Hint: Write yi-y as yi-μ- σ-μ), verify that r and the population variance σ 17 and cither take espectation over all pouible samples or define an indicator variable for each unit, indicating whether it is included in the sample.]
For purposes of studying sampling distribution, we consider a small population of N = 4 units, labeled 1, 2, 3, 4, with respective y-values yı = 3, y2 = 1, y3 = 0, y4 = 5. (c) Plan 2: Consider a simple random sample with replacement (SRSWR) design with sample size n=2. (i) Find the number of possible SRSs of size n = 2. List every possible sample. For each sample, what is the probability that it is the one...
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...
I1. Follow the steps below to show that the pooled estimator $p is an unbi- ased estimator for the common standard deviation of two independent sam ples Let Yi, Yi2, ..., Yini denote the random sample of size n from the first population with population mean μ| and population variance σ, and let Y21, Y22, ..., Y2na denote an independent random sample of size n2 from the second population with population mean μ2 and population mean ơ3. Sup- pose that...
CLUSTER SAMPLING WITH ESTIMATION Suppose a population of size N is divided into K- N/M groups of size M. We select a sample of size n -km the following way: » First we select k groups out of K groups by simple random sampling . We then select m units in each group selected on the first step by simple random sampling . The estimate of the population mean is the average Y of the sample. Let μί be the...
Assume a population of 1, 4, and 10. Assume that samples of size n = 2 are randomly selected with replacement from the population. Listed below are the nine different samples. Complete parts a through d below. 1,1 1,4 1,10 4,1 4,4 4,10 10,1 10,4 10,10 o a. Find the value of the population standard deviation o. (Round to three decimal places as needed.) b. Find the standard deviation of each of the nine samples, then summarize the sampling distribution...
4
and 5
samples, the other in small samples. Which is which? Explain. (d) Suppose we know that the 5 values are from a symmetric distribution. Then the sample median is also unbiased and consistent for the population mean. The sample mean has lower variance. Would you prefer to use the sample 4. Suppose Yi, Y, are iid r ables with E(n)-μ, Var(K)-σ2 < oo. For large n, find the approximate 5. Suppose we observe Yi...Yn from a normal distribution...
Assume a population of 2, 4, and 9. Assume that samples of size n 2 are randomly selected with replacement from the population. Listed below are the nine different samples. Complete parts a through d below 4.9 9,9 2,2 2,4 2,9 4,4 9,2 9.4 a. Find the value of the population standard deviation σ (Round to three decimal places as needed.) b. Find the standard deviation of each of the nine samples, then summarize the sampling distribution of the standard...
Please give detailed steps. Thank you.
5. Let {X1, X2,..., Xn) denote a random sample of size N from a population d escribed by a random variable X. Let's denote the population mean of X by E(X) - u and its variance by Consider the following four estimators of the population mean μ : 3 (this is an example of an average using only part of the sample the last 3 observations) (this is an example of a weighted average)...
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