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5 Consider the population described by the probabllty tribution shown here: 1 3 4. p(x) .2 .3 .2 .2 .1 The random variable x is observed twice. If these observa- tions are independent, verify that the different samples of size 2 and their probabilities are as follows:

5 Consider the population described by the probabllty tribution shown here: 1 3 4. p(x) .2 .3 .2 .2 .1 The random variable x is observed twice. If these observa- tions are independent, verify that the different samples of size 2 and their probabilities are as follows:

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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### Sampling Distributions As noted earlier, many sampling distributions can be derived mathematically, but the theory necessary to do so is beyond the scope of this text. Consequently, when we need to know the properties of a statistic, we will present its sampling distribution and verbally describe its properties. Several of the important properties we look for in sampling distributions are discussed in the next section. --- **Exercise 6.5** Consider the population described by the probability distribution shown here: | x | 1 | 2 | 3 | 4 | 5 | |-----|-----|-----|-----|-----|-----| | p(x)| .2 | .3 | .2 | .2 | .1 | The random variable \( x \) is observed twice. If these observations are independent, verify that the different samples of size 2 and their probabilities are as follows: | Sample | Probability | Sample | Probability | |--------|-------------|--------|-------------| | 1, 1 | .04 | 3, 4 | .04 | | 1, 2 | .06 | 3, 5 | .02 | | 1, 3 | .04 | 4, 4 | .04 | | 1, 4 | .04 | 4, 5 | .02 | | 1, 5 | .02 | 5, 5 | .01 | | 2, 1 | .06 | 2, 2 | .09 | | 2, 3 | .06 | 2, 4 | .06 | | 2, 5 | .03 | | | **Tasks:** a. Find the sampling distribution of the sample mean \( \bar{x} \). b. Construct a probability histogram for the sampling distribution of \( \bar{x} \). c. What is the probability that \( \bar{x} \) is 4.5 or larger? d. Would you expect to observe a value of \( \bar{x} \) equal to 4.5 or larger? Explain. --- **Exercise 6.6** Refer to Exercise 6.5 and find \( E(x) = \mu \
Transcribed Image Text:### Sampling Distributions As noted earlier, many sampling distributions can be derived mathematically, but the theory necessary to do so is beyond the scope of this text. Consequently, when we need to know the properties of a statistic, we will present its sampling distribution and verbally describe its properties. Several of the important properties we look for in sampling distributions are discussed in the next section. --- **Exercise 6.5** Consider the population described by the probability distribution shown here: | x | 1 | 2 | 3 | 4 | 5 | |-----|-----|-----|-----|-----|-----| | p(x)| .2 | .3 | .2 | .2 | .1 | The random variable \( x \) is observed twice. If these observations are independent, verify that the different samples of size 2 and their probabilities are as follows: | Sample | Probability | Sample | Probability | |--------|-------------|--------|-------------| | 1, 1 | .04 | 3, 4 | .04 | | 1, 2 | .06 | 3, 5 | .02 | | 1, 3 | .04 | 4, 4 | .04 | | 1, 4 | .04 | 4, 5 | .02 | | 1, 5 | .02 | 5, 5 | .01 | | 2, 1 | .06 | 2, 2 | .09 | | 2, 3 | .06 | 2, 4 | .06 | | 2, 5 | .03 | | | **Tasks:** a. Find the sampling distribution of the sample mean \( \bar{x} \). b. Construct a probability histogram for the sampling distribution of \( \bar{x} \). c. What is the probability that \( \bar{x} \) is 4.5 or larger? d. Would you expect to observe a value of \( \bar{x} \) equal to 4.5 or larger? Explain. --- **Exercise 6.6** Refer to Exercise 6.5 and find \( E(x) = \mu \
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