Final answer:
To achieve a mean of 28 and a mode of 15 with the data set, we must add two more instances of 15 and a value of 2. The missing values in the data set are thus 15 and 2.
Step-by-step explanation:
To find the missing values in the data so that the mean is 28, and the mode is 15, we first need to understand what these terms mean. The mean is the average of all numbers in the data set, and the mode is the number that occurs most frequently.
First, we need to calculate the sum of the given numbers: 20 + 40 + 36 + 15 + 38 + 30 + 41 = 220.
Since the mean is 28, the total sum of all the numbers in the data set should be 28 times the number of values. If we denote the missing values as x and y, we get: 220 + x + y = 28 * (7 + 2). Simplifying, we find that x + y = 56.
Given that the mode is 15, and mode is the most frequently occurring number, we need at least one more 15 to make it the mode. Thus, x = 15. Now we can solve for y: 15 + y = 56, which results in y being 41.
However, having a second 41 would make it a second mode, violating the requirement of 15 being the only mode. Therefore, this initial solution is not viable.
Let's reassess our approach. To guarantee 15 as the only mode, we must introduce another 15, making two more instances of it in the dataset. Now our equation adjusting for two additional 15's looks like this: 220 + 15 + 15 = 28 * (7 + 2). Solving for y, we get: 250 + y = 252, leading to y = 2.
Therefore, the missing values that satisfy the conditions for the mean and the mode are 15 (to account for the mode) and 2 (to ensure the mean remains 28).