Question 1177077
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How many different samples (without replacement) do we have in this case?<br>
ANSWER: "5 choose 2" = (5*4)/(2*1) = 20/2 = 10<br>
How many different values does the random variable x̄ assume?<br>
To this point in your post you have not specifically that x-bar is the average.  But since you talk about the average later in your post, and since using x-bar for an average is standard, I will assume that is what it is.<br>
The smallest possible sum is 1+2=3 and the largest is 4+5=9, and clearly all the whole number sums between 3 and 9 can be obtained.  That make seven different values for the sum; and that of course means seven different values for the average.
ANSWER: 7<br>
What is the probability of getting average 5?<br>
The largest sum we can get is 9; the largest average we can get is 4.5.<br>
ANSWER: P(x-bar=5) = 0<br>
What is the probability of getting average 2.5?
An average of 2.5 means a sum of 5.  There are 2 ways to get a sum of 5 -- 1 and 4, or 2 and 3.  Since there are 10 different samples possible....<br>
ANSWER: P(x-bar=2.5) = 2/10 = 1/5<br>