1. **State the problem:** We have a population of 8 cards numbered 1 to 8. We draw samples of size 3 without replacement and want to find the sampling distribution of the sample means. 2. **Formula and rules:** The sample mean for a sample $\{x_1, x_2, x_3\}$ is given by: $$\bar{x} = \frac{x_1 + x_2 + x_3}{3}$$ We must consider all possible combinations of 3 cards from 8, which is $\binom{8}{3} = 56$ samples. 3. **List all combinations and calculate their means:** The samples are all 3-card combinations from \{1,2,3,4,5,6,7,8\}. 4. **Calculate sample means for each combination:** For example, sample \{1,2,3\} mean is $\frac{1+2+3}{3} = 2$. 5. **Find unique sample means and their frequencies:** Possible means range from $\frac{1+2+3}{3}=2$ up to $\frac{6+7+8}{3}=7$. 6. **Construct the sampling distribution:** Calculate the frequency of each mean and divide by 56 to get probabilities. 7. **Summary of sampling distribution (mean : frequency):** - 2.0 : 1 - 2.333 : 2 - 2.667 : 2 - 3.0 : 3 - 3.333 : 4 - 3.667 : 4 - 4.0 : 5 - 4.333 : 5 - 4.667 : 5 - 5.0 : 5 - 5.333 : 4 - 5.667 : 4 - 6.0 : 3 - 6.333 : 2 - 6.667 : 2 - 7.0 : 1 Each probability is frequency divided by 56. This distribution shows the probabilities of each sample mean when drawing 3 cards from 8 without replacement.