Question
71 + 91 + 82 + 90 + 88 + 61 + 70 = 553 What is the missing data set that would make the mean equal 80? Show all steps
Answers
GPT-4o mini
To find the missing data set that would make the mean equal to 80, we first need to calculate how many data values we have and how their sum can help us achieve a mean of 80.
### Steps:
1. **Calculate the current total and the data count**:
- The current dataset is: 71, 91, 82, 90, 88, 61, 70.
- The sum of these numbers is already calculated as 553.
- The count (n) of these numbers = 7.
2. **Define the goal**:
- We want the mean of the new dataset (including the missing data set) to be 80.
3. **Let "x" be the number of additional data points added**:
- We'll denote the total number of values after adding the missing data as \( n + x = 7 + x \).
4. **Define the total sum needed**:
- The mean formula is:
\[
\text{Mean} = \frac{\text{Total sum}}{\text{Total count}}
\]
- To make the mean 80:
\[
80 = \frac{553 + S}{7 + x}
\]
where \( S \) is the sum of the additional values.
5. **Rearranging the equation**:
- Rearranging gives:
\[
80(7 + x) = 553 + S
\]
\[
560 + 80x = 553 + S
\]
\[
S = 80x + 7
\]
6. **Exploring values for x**:
- The value of \( S \) (the sum of the additional data values) depends on how many additional values \( x \) we decide to add.
### Option 1: Adding One Value (x = 1)
- If we set \( x = 1 \):
\[
S = 80(1) + 7 = 87.
\]
### Option 2: Adding Two Values (x = 2)
- If we set \( x = 2 \):
\[
S = 80(2) + 7 = 167.
\]
### General Conclusion:
In general, we can calculate this for any number \( x \).
**For specific examples:**
- Adding one additional value of 87 would lead to:
\[
S = 553 + 87 = 640 \quad \text{and} \quad 8 \text{ data points, thus} \quad \text{Mean} = \frac{640}{8} = 80.
\]
Thus, the additional data points required depend on how many you choose to add, but they must sum to a specific value as calculated, \( 80x + 7 \), to achieve an overall mean of 80.
For example, adding just (87) keeps the mean correct, but with more data points, the values must be adjusted accordingly.
### Steps:
1. **Calculate the current total and the data count**:
- The current dataset is: 71, 91, 82, 90, 88, 61, 70.
- The sum of these numbers is already calculated as 553.
- The count (n) of these numbers = 7.
2. **Define the goal**:
- We want the mean of the new dataset (including the missing data set) to be 80.
3. **Let "x" be the number of additional data points added**:
- We'll denote the total number of values after adding the missing data as \( n + x = 7 + x \).
4. **Define the total sum needed**:
- The mean formula is:
\[
\text{Mean} = \frac{\text{Total sum}}{\text{Total count}}
\]
- To make the mean 80:
\[
80 = \frac{553 + S}{7 + x}
\]
where \( S \) is the sum of the additional values.
5. **Rearranging the equation**:
- Rearranging gives:
\[
80(7 + x) = 553 + S
\]
\[
560 + 80x = 553 + S
\]
\[
S = 80x + 7
\]
6. **Exploring values for x**:
- The value of \( S \) (the sum of the additional data values) depends on how many additional values \( x \) we decide to add.
### Option 1: Adding One Value (x = 1)
- If we set \( x = 1 \):
\[
S = 80(1) + 7 = 87.
\]
### Option 2: Adding Two Values (x = 2)
- If we set \( x = 2 \):
\[
S = 80(2) + 7 = 167.
\]
### General Conclusion:
In general, we can calculate this for any number \( x \).
**For specific examples:**
- Adding one additional value of 87 would lead to:
\[
S = 553 + 87 = 640 \quad \text{and} \quad 8 \text{ data points, thus} \quad \text{Mean} = \frac{640}{8} = 80.
\]
Thus, the additional data points required depend on how many you choose to add, but they must sum to a specific value as calculated, \( 80x + 7 \), to achieve an overall mean of 80.
For example, adding just (87) keeps the mean correct, but with more data points, the values must be adjusted accordingly.