Question

Random samples of size 3 are taken from a population of the numbers 3, 4, 5, 6, 7, 8, and 9.

1. How many samples are possible? 

2. Construct the sampling distribution of the sample means. 

Answer 1

1. Consider all samples with replacement. We get: $7^3=343$ possible samples. For samples without replacement we get: $7\cdot6\cdot5=210.$
2. We consider sampling distribution of the sample means without replacements. We receive samples of the type: $(i,j,k),$ where $i\neq j\neq k$. The probability of the mean $\frac{i+j+k}{3}$ is: $\frac{1}{210}\cdot N$, where $N$ is a number of samples with the same mean. The number can be found either from the table with samples or from equality $i_1+j_1+k_1=i_2+j_2+k_2$ that has to be considered for all samples.