Let's break down the problem step by step.

We are given the following information:

• The data set has 8 positive integers greater than 10.
• Seven of these integers are: 21, 24, 25, 19, 24, 18, 23.
• The mean of these 7 integers is 22.
• The mean of the full data set (all 8 integers) is an integer less than 22.

We need to find the value of the smallest integer in the full data set.

Step 1: Calculate the sum of the 7 integers.

The mean of the 7 integers is 22, so we can calculate their sum using the formula:

$\text{Mean} = \frac{\text{Sum of the numbers}}{\text{Number of the numbers}}$

So,

$22 = \frac{\text{Sum of the 7 integers}}{7}$

$\text{Sum of the 7 integers} = 22 \times 7 = 154$

Step 2: Let the missing 8th integer be $x$.

Now, the sum of all 8 integers will be:

$\text{Sum of all 8 integers} = 154 + x$

Step 3: Set up the mean of the full data set.

The mean of the full data set (8 numbers) is less than 22. Using the formula for the mean again:

$\text{Mean of all 8 numbers} = \frac{154 + x}{8}$

We are told this mean is less than 22:

$\frac{154 + x}{8} < 22$

Step 4: Solve the inequality.

First, multiply both sides of the inequality by 8:

$154 + x < 176$

Now, subtract 154 from both sides:

$x < 22$

Step 5: Check the values.

We are also told that all integers are greater than 10, so $x$ must be between 10 and 22. The integers in this range are 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and 21.

Among these, 14 is the only one that keeps the overall mean less than 22. Let's check:

If $x = 14$, the sum of all 8 integers becomes:

$154 + 14 = 168$

The mean of all 8 integers is:

$\frac{168}{8} = 21$

Since the mean is less than 22, this satisfies the condition.

Answer:

The smallest integer in the full data set is 14. Thus, the correct answer is B.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How would the solution change if the mean of the 8 integers was exactly 22?
2. What if the 7 given integers were all different? How would that affect the solution?
3. Can you find other possible values of the smallest integer if the mean condition was changed to "less than 21"?
4. How would this problem change if there were 9 integers instead of 8?
5. Can you derive a general formula to find the missing integer in a similar problem?

Tip: When working with means, always double-check your sums and remember that multiplying or dividing inequalities can flip the inequality sign!