Consider a population consisting of 5 elements: {1,2,3,4,5}. We will draw a sample of size two from this population and calculate the average of drawn numbers. For example, we can get numbers 1 and 5, the average is then 3 or numbers 3 and 4, their average is 3.5. The averages with their probabilities are probability distribution. It is an example of a sampling distribution with sample size 2.

Question

How many different samples (with replacement) we have in this case?

How many different values does the random variable x̄ assume?

What is the probability of getting average 5?

What is the probability of getting average 2.5?

(write your answer as a fraction a/b or as an integer)

Answer 1

1. How many different samples (with replacement) do we have in this case?

Since we are sampling with replacement and the sample size is 2, each of the two selections can be any of the 5 elements from the population. Therefore, the total number of different samples is:

\[ 5 \times 5 = 25 \]

2. How many different values does the random variable \( \bar{x} \) assume?

To find the different values the average \( \bar{x} \) can take, we need to calculate the averages for all possible pairs of numbers in the population {1, 2, 3, 4, 5}:

• Pair (1, 1): \( \frac{1 + 1}{2} = 1 \)

• Pair (1, 2): \( \frac{1 + 2}{2} = 1.5 \)

• Pair (1, 3): \( \frac{1 + 3}{2} = 2 \)

• Pair (1, 4): \( \frac{1 + 4}{2} = 2.5 \)

• Pair (1, 5): \( \frac{1 + 5}{2} = 3 \)

• Pair (2, 2): \( \frac{2 + 2}{2} = 2 \)

• Pair (2, 3): \( \frac{2 + 3}{2} = 2.5 \)

• Pair (2, 4): \( \frac{2 + 4}{2} = 3 \)

• Pair (2, 5): \( \frac{2 + 5}{2} = 3.5 \)

• Pair (3, 3): \( \frac{3 + 3}{2} = 3 \)

• Pair (3, 4): \( \frac{3 + 4}{2} = 3.5 \)

• Pair (3, 5): \( \frac{3 + 5}{2} = 4 \)

• Pair (4, 4): \( \frac{4 + 4}{2} = 4 \)

• Pair (4, 5): \( \frac{4 + 5}{2} = 4.5 \)

• Pair (5, 5): \( \frac{5 + 5}{2} = 5 \)

From this, the unique average values that the random variable \( \bar{x} \) can assume are:

• 1
• 1.5
• 2
• 2.5
• 3
• 3.5
• 4
• 4.5
• 5

Thus, the different values that \( \bar{x} \) can assume are 9.

3. What is the probability of getting an average of 5?

To achieve an average of 5, the only valid sample is (5, 5):

• There is 1 sample that results in \( \bar{x} = 5 \).

The total number of samples is 25, therefore:

\[ P(\bar{x} = 5) = \frac{1}{25} \]

4. What is the probability of getting an average of 2.5?

To achieve an average of 2.5, the valid samples are:

• (1, 4)
• (4, 1)
• (2, 3)
• (3, 2)

Thus, there are 4 samples that result in \( \bar{x} = 2.5 \).

So the probability is:

\[ P(\bar{x} = 2.5) = \frac{4}{25} \]

Summary of Answers:

• Different samples (with replacement): \( 25 \)
• Different values \( \bar{x} \) assumes: \( 9 \)
• Probability of average 5: \( \frac{1}{25} \)
• Probability of average 2.5: \( \frac{4}{25} \)