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, 1 | .04 |
1, 4 | .04 | 4, 2 | .06 |
1, 5 | .02 | 4, 3 | .04 |
2, 1 | .06 | 4, 4 | .04 |
2, 2 | .09 | 4, 5 | .02 |
2, 3 | .06 | 5, 1 | .02 |
2, 4 | .06 | 5, 2 | .03 |
2, 5 | .03 | 5, 3 | .02 |
3, 1 | .04 | 5, 4 | .02 |
3, 2 | .06 | 5, 5 | .01 |
3, 3 | .04 |
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a. Find the sampling distribution of the sample mean x.
b. Construct a probability histogram for the sampling distribution of x.
c. What is the probability that x is 4.5 or larger?
d. Would you expect to observe a value of x equal to 4.5 or larger? Explain.
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