Starting with the data values 70 and 100, add three data values to the sample so that the mean is 96, the median is 97, and the mode is 97. Please show all of your work
For the mode = 97, at least 2 scores need to = 97
Let x = score
(x + 97 + 97 + 70 + 100)/5 = 96
x + 97 + 97 + 70 + 100 = 480
x = 116
With these scores
70, 97, 97, 100, 116, what would the median be?
To find the three additional values that satisfy the given conditions, we can follow these steps:
1. Start by calculating the sum of the two initial data values:
Sum = 70 + 100 = 170
2. Since we need to find the mean, we divide the sum by the total number of values (including the additional three values) to get the desired mean of 96:
Mean = 96
Total number of values = 2 + 3 = 5
Sum / Total number of values = 96
170 / 5 = 96
34 = 96 - 170
3. We subtract the sum we already had from the desired sum (96 * 5 = 480) to get the sum of the additional three values:
Sum of additional three values = 480 - 170 = 310
4. Next, we know that the median is 97, which means that the middle value (when the values are arranged in ascending order) should be 97. Since we already have two values (70 and 100), we need to add three more. So, the third value (the median) should be 97.
5. Now, we need to determine the remaining two values such that the mean is 96, and the mode is 97. Since the mode is the value that appears most frequently, we should have at least two 97s in the dataset.
Let's denote the remaining two values as x and y.
The sum of the five values can be written as: 70 + 100 + 97 + x + y = 480.
6. We know that the mean is calculated by dividing the sum by the number of values. So, we can write:
(70 + 100 + 97 + x + y) / 5 = 96
7. Multiplying both sides of the equation by 5 yields:
267 + x + y = 480
8. Rearranging the equation gives:
x + y = 480 - 267
x + y = 213
9. Since the mode is 97, we want to ensure that 97 appears most frequently. Let's assume that x = y = 97.
If x and y are both 97, then the equation becomes:
97 + 97 = 213
194 ≠ 213.
10. Since x + y should be equal to 213, we can try different values for x and y, making sure that 97 is one of them. Let's try x = 87 and y = 126:
87 + 126 = 213
213 = 213
The equation is satisfied.
11. Therefore, the three additional values that satisfy the given conditions are: 87, 97, and 126.
In summary, the five values in the dataset are: 70, 100, 87, 97, and 126.