Question

question number 9. (10.00 points) a random sample of 2 measurements is taken from the following population of values: -2, -1, 1, 2, 5. what is the probability that the range of the sample is 1? 0.5 0.3 0.4 0.2 0.1 none of the above

Explanation:

Step1: List all possible samples

The population values are - 2, - 1, 1, 2, 5. The number of ways to choose a sample of size \(n = 2\) from a population of size \(N=5\) is given by the combination formula \(C(N,n)=\frac{N!}{n!(N - n)!}\), but we can also list them out. The possible samples are \((-2,-1),(-2,1),(-2,2),(-2,5),(-1,1),(-1,2),(-1,5),(1,2),(1,5),(2,5)\) and their reverse - ordered pairs (since the order of selection in a sample matters in terms of calculating range).

Step2: Calculate the range for each sample

The range of a sample is defined as the difference between the maximum and minimum values in the sample.
For sample \((-2,-1)\), range \(=|-1-(-2)| = 1\); for sample \((-1,-2)\), range \(=|-2 - (-1)|=1\).
For sample \((-2,1)\), range \(=|1-(-2)| = 3\); for sample \((1,-2)\), range \(=|-2 - 1|=3\).
For sample \((-2,2)\), range \(=|2-(-2)| = 4\); for sample \((2,-2)\), range \(=|-2 - 2|=4\).
For sample \((-2,5)\), range \(=|5-(-2)| = 7\); for sample \((5,-2)\), range \(=|-2 - 5|=7\).
For sample \((-1,1)\), range \(=|1-(-1)| = 2\); for sample \((1,-1)\), range \(=|-1 - 1|=2\).
For sample \((-1,2)\), range \(=|2-(-1)| = 3\); for sample \((2,-1)\), range \(=|-1 - 2|=3\).
For sample \((-1,5)\), range \(=|5-(-1)| = 6\); for sample \((5,-1)\), range \(=|-1 - 5|=6\).
For sample \((1,2)\), range \(=|2 - 1|=1\); for sample \((2,1)\), range \(=|1 - 2|=1\).
For sample \((1,5)\), range \(=|5 - 1|=4\); for sample \((5,1)\), range \(=|1 - 5|=4\).
For sample \((2,5)\), range \(=|5 - 2|=3\); for sample \((5,2)\), range \(=|2 - 5|=3\).
The total number of samples of size 2 from a population of size 5 is \(n(S)=5\times4 = 20\) (using the multiplication principle, as for the first element of the sample we have 5 choices and for the second element we have 4 choices).
The number of samples with range 1 is 4: \((-2,-1),(-1,-2),(1,2),(2,1)\).

Step3: Calculate the probability

The probability \(P\) that the range of the sample is 1 is given by the formula \(P=\frac{n(A)}{n(S)}\), where \(n(A)\) is the number of favorable outcomes and \(n(S)\) is the total number of outcomes.
\(P=\frac{4}{20}=0.2\)

Answer:

0.2