Assume a population of any 4 numbers(suppose: 4,8,15,19). Select a random sample of size 2 with replacement from the population. List all the possible samples and their corresponding values of x̄ “x-bar” and then find the probability distribution of “X-bar”
study the following:
The distribution of X and the sampling distribution of X-bar
To begin, let's first understand the population and the concept of a sample with replacement.
The given population is: 4, 8, 15, 19
A sample with replacement means that after selecting a number from the population, it is put back in, and the population remains the same for the next selection.
Now, let's find all the possible samples of size 2 that can be taken from this population:
Sample 1: (4, 4)
Sample 2: (4, 8)
Sample 3: (4, 15)
Sample 4: (4, 19)
Sample 5: (8, 4)
Sample 6: (8, 8)
Sample 7: (8, 15)
Sample 8: (8, 19)
Sample 9: (15, 4)
Sample 10: (15, 8)
Sample 11: (15, 15)
Sample 12: (15, 19)
Sample 13: (19, 4)
Sample 14: (19, 8)
Sample 15: (19, 15)
Sample 16: (19, 19)
Now, let's calculate the sample means (x-bar) for each of these samples:
Sample 1: x-bar = (4 + 4) / 2 = 4
Sample 2: x-bar = (4 + 8) / 2 = 6
Sample 3: x-bar = (4 + 15) / 2 = 9.5
Sample 4: x-bar = (4 + 19) / 2 = 11.5
Sample 5: x-bar = (8 + 4) / 2 = 6
Sample 6: x-bar = (8 + 8) / 2 = 8
Sample 7: x-bar = (8 + 15) / 2 = 11.5
Sample 8: x-bar = (8 + 19) / 2 = 13.5
Sample 9: x-bar = (15 + 4) / 2 = 9.5
Sample 10: x-bar = (15 + 8) / 2 = 11.5
Sample 11: x-bar = (15 + 15) / 2 = 15
Sample 12: x-bar = (15 + 19) / 2 = 17
Sample 13: x-bar = (19 + 4) / 2 = 11.5
Sample 14: x-bar = (19 + 8) / 2 = 13.5
Sample 15: x-bar = (19 + 15) / 2 = 17
Sample 16: x-bar = (19 + 19) / 2 = 19
Now, let's determine the probability distribution of X-bar.
The probability distribution is determined by calculating the frequency (count) of each unique x-bar value and dividing it by the total number of samples.
In this case, each x-bar value occurs either once or multiple times, and we have a total of 16 samples.
Using the above results, we can construct the following probability distribution:
x-bar value: 4, Probability: 1/16
x-bar value: 6, Probability: 1/16
x-bar value: 9.5, Probability: 2/16
x-bar value: 11.5, Probability: 3/16
x-bar value: 13.5, Probability: 2/16
x-bar value: 15, Probability: 1/16
x-bar value: 17, Probability: 2/16
x-bar value: 19, Probability: 1/16
So, this is the probability distribution of X-bar for the given population and sample size of 2 with replacement.