Question

What is the missing value in the data set that would make the mean equal 80? Show your work or explain how you got your answer. {71, 91, 82, 90, 88, 61, 70, __ }

Answer 1

To find the missing value in the dataset that would make the mean equal to 80, we can follow these steps:

1. **Understand the formula for the mean**: The mean of a dataset is calculated by taking the sum of all the values and dividing it by the number of values.

\[
\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}}
\]

2. **Set up the equation for the mean**: In this dataset, we have seven known values and one missing value (let's call it \( x \)). Thus, the total number of values is 8. We want the mean to equal 80.

\[
80 = \frac{71 + 91 + 82 + 90 + 88 + 61 + 70 + x}{8}
\]

3. **Calculate the sum of the known values**:

\[
71 + 91 + 82 + 90 + 88 + 61 + 70 = 453
\]

4. **Substitute and solve for \( x \)**:

Plug the sum into the equation:

\[
80 = \frac{453 + x}{8}
\]

Multiply both sides by 8 to eliminate the fraction:

\[
640 = 453 + x
\]

Now, solve for \( x \):

\[
x = 640 - 453 = 187
\]

5. **Conclusion**: The missing value needed to make the mean equal 80 is \( \boxed{187} \).

To verify, you can add \( 187 \) to the sum of the other values and calculate the new mean:

\[
\text{New Sum} = 453 + 187 = 640
\]
\[
\text{Mean} = \frac{640}{8} = 80
\]

This confirms that the calculation is correct.