1. **State the problem:** We have 7 numbers with a mean of 4. Six of the numbers are 2, 3, 1, 2, 7, and 8. We need to find: a) The seventh number. b) The median of all 7 numbers. 2. **Formula for mean:** The mean of $n$ numbers $x_1, x_2, ..., x_n$ is given by $$\text{mean} = \frac{x_1 + x_2 + ... + x_n}{n}$$ 3. **Find the sum of all 7 numbers:** Since the mean is 4 and there are 7 numbers, $$\text{sum} = \text{mean} \times n = 4 \times 7 = 28$$ 4. **Sum of the known six numbers:** $$2 + 3 + 1 + 2 + 7 + 8 = 23$$ 5. **Find the seventh number $x$:** $$23 + x = 28$$ $$x = 28 - 23 = 5$$ 6. **List all 7 numbers including the seventh:** $$\{1, 2, 2, 3, 5, 7, 8\}$$ 7. **Find the median:** The median is the middle number when the numbers are sorted. Since there are 7 numbers (odd), the median is the 4th number in the sorted list. The sorted list is already given above. The 4th number is $3$. **Final answers:** - a) The seventh number is $5$. - b) The median of the numbers is $3$.