Question

question number 9. (10.00 points) a random sample of 2 measurements is taken from the following population of values: 1, 2, 4, 5, 8. what is the probability that the range of the sample is 3? 0.5 0.4 0.6 0.1 0.3 none of the above

Explanation:

Step1: List all possible samples of size 2

The population is {1, 2, 4, 5, 8}. The number of possible samples of size \(n = 2\) using the combination formula \(C(N,n)=\frac{N!}{n!(N - n)!}\), where \(N=5\) and \(n = 2\), is \(C(5,2)=\frac{5!}{2!(5 - 2)!}=\frac{5\times4}{2\times1}=10\). The 10 samples are \((1,2),(1,4),(1,5),(1,8),(2,4),(2,5),(2,8),(4,5),(4,8),(5,8)\).

Step2: Calculate the range for each sample

The range of a sample \((x_1,x_2)\) is \(R=\max(x_1,x_2)-\min(x_1,x_2)\).

• For \((1,2)\), \(R = 2 - 1=1\).
• For \((1,4)\), \(R = 4 - 1 = 3\).
• For \((1,5)\), \(R = 5 - 1=4\).
• For \((1,8)\), \(R = 8 - 1 = 7\).
• For \((2,4)\), \(R = 4 - 2=2\).
• For \((2,5)\), \(R = 5 - 2 = 3\).
• For \((2,8)\), \(R = 8 - 2=6\).
• For \((4,5)\), \(R = 5 - 4 = 1\).
• For \((4,8)\), \(R = 8 - 4=4\).
• For \((5,8)\), \(R = 8 - 5 = 3\).

Step3: Calculate the probability

The number of samples with range equal to 3 is 3. The probability \(P\) that the range of the sample is 3 is the number of favorable outcomes divided by the total number of outcomes. So \(P=\frac{3}{10}=0.3\).

Answer:

0.3