There are 120 possible samples of size n=3 which can be drawn from this population of 10 scores: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Write each sample and calculate its mean
To calculate the mean of each sample, we need to start by finding all possible combinations of size 3 from the population of 10 scores: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
To generate all possible combinations, we can use a mathematical concept called combinations or binomial coefficients. The formula to calculate the number of combinations is:
C(n, r) = n! / (r! * (n - r)!),
where n is the total number of elements, r is the number of elements in each combination, and ! denotes factorial.
In our case, n = 10 (the number of elements in the population) and r = 3 (the sample size we want).
Plugging the values into the formula, we have:
C(10, 3) = 10! / (3! * (10 - 3)!)
= 10! / (3! * 7!)
Simplifying further:
10! = 10 * 9 * 8 * 7!
3! = 3 * 2 * 1
C(10, 3) = (10 * 9 * 8 * 7!) / (3 * 2 * 1 * 7!)
= (10 * 9 * 8) / (3 * 2 * 1)
= 120
Therefore, there are 120 possible samples of size 3 that can be drawn from this population.
Now let's write each sample and calculate its mean:
1. Sample: 3, 4, 5
Mean: (3 + 4 + 5) / 3 = 4
2. Sample: 3, 4, 6
Mean: (3 + 4 + 6) / 3 = 4.333...
3. Sample: 3, 4, 7
Mean: (3 + 4 + 7) / 3 = 4.666...
4. Sample: 3, 4, 8
Mean: (3 + 4 + 8) / 3 = 5
5. Sample: 3, 4, 9
Mean: (3 + 4 + 9) / 3 = 5.333...
6. Sample: 3, 4, 10
Mean: (3 + 4 + 10) / 3 = 5.666...
7. Sample: 3, 4, 11
Mean: (3 + 4 + 11) / 3 = 6
8. Sample: 3, 4, 12
Mean: (3 + 4 + 12) / 3 = 6.333...
9. Sample: 3, 5, 6
Mean: (3 + 5 + 6) / 3 = 4.666...
10. Sample: 3, 5, 7
Mean: (3 + 5 + 7) / 3 = 5
...continue with the rest of the samples until all 120 samples have been calculated.