Question

the ages (years) of three government officials when they died in office were 54, 45, and 61. complete parts (a) through (d).
a. assuming that 2 of the ages are randomly selected with replacement, list the different possible samples.
a. (54,45),(54,61),(45,54),(45,61),(61,54),(61,45)
b. (54,54), (54,45),(54,61),(45,54),(45,45),(45,61),(61,54),(61,45),(61,61)
c. (54,54), (54,45),(54,61),(45,45),(45,61),(61,61)
d. (54,45),(54,61),(45,61)
b. find the range of each of the samples, then summarize the sampling distribution of the ranges in the format of a table representing the probability distribution.
sample range probability
(type an integer or a fraction.)

Explanation:

Step1: Calculate the range for each sample in option B

The range is calculated as \(Range = max - min\).

• For \((54,54)\): \(Range=54 - 54=0\)
• For \((54,45)\): \(Range = 54-45 = 9\)
• For \((54,61)\): \(Range=61 - 54=7\)
• For \((45,54)\): \(Range=54 - 45=9\)
• For \((45,45)\): \(Range=45 - 45=0\)
• For \((45,61)\): \(Range=61 - 45 = 16\)
• For \((61,54)\): \(Range=61 - 54=7\)
• For \((61,45)\): \(Range=61 - 45=16\)
• For \((61,61)\): \(Range=61 - 61=0\)

Step2: Count the frequency of each range

• Range \(0\): occurs \(3\) times (\((54,54)\), \((45,45)\), \((61,61)\))
• Range \(7\): occurs \(2\) times (\((54,61)\), \((61,54)\))
• Range \(9\): occurs \(2\) times (\((54,45)\), \((45,54)\))
• Range \(16\): occurs \(2\) times (\((45,61)\), \((61,45)\))

Step3: Calculate the probability for each range

The total number of samples \(n = 9\).

• Probability of range \(0\): \(P(0)=\frac{3}{9}=\frac{1}{3}\)
• Probability of range \(7\): \(P(7)=\frac{2}{9}\)
• Probability of range \(9\): \(P(9)=\frac{2}{9}\)
• Probability of range \(16\): \(P(16)=\frac{2}{9}\)

Answer:

Sample Range	Probability
\(7\)	\(\frac{2}{9}\)
\(9\)	\(\frac{2}{9}\)
\(16\)	\(\frac{2}{9}\)