Greg teaches an art class. The table below shows how many drawings his students had submitted by last Friday. Greg calculates the mean absolute deviation (MAD) of the data. Then, one student submits 25 additional drawings. Greg cannot remember whether the drawings are Amy's or Emily's, but he thinks the MAD will increase no matter who submitted the drawings. Is Greg correct? Use the drop-down menus to explain your reasoning.

Greg's Students' Drawings

\begin{tabular}{|c|c|}
\hline
Student & Number of Drawings Submitted \\
\hline
Amy & 6 \\
\hline
Bob & 34 \\
\hline
Christa & 35 \\
\hline
Diego & 37 \\
\hline
Emily & 43 \\
\hline
\end{tabular}

Click the arrows to choose an answer from each menu.

The MAD of the data in the table is [tex]\(\square\)[/tex] .

If the additional drawings are Amy's, the MAD of the data set will [tex]\(\square\)[/tex] .

If they are Emily's, the MAD will [tex]\(\square\)[/tex] .

The MAD of the new data set [tex]\(\square\)[/tex] depend on whether it was Amy or Emily who turned in the additional drawings.

So, Greg is [tex]\(\square\)[/tex] .



Answer :

To analyze whether Greg's assumption about the Mean Absolute Deviation (MAD) is correct, let's go through the provided data and calculate the needed values step-by-step.

### Step 1: Initial Data Set

The initial number of drawings submitted by each student are:
- Amy: 6
- Bob: 34
- Christa: 35
- Diego: 37
- Emily: 43

So, the initial data set is: [tex]\([6, 34, 35, 37, 43]\)[/tex]

### Step 2: Calculate the Initial Mean

The mean of the initial set of drawings:
[tex]\[ \text{mean} = \frac{6 + 34 + 35 + 37 + 43}{5} = \frac{155}{5} = 31 \][/tex]

### Step 3: Calculate the Initial Mean Absolute Deviation (MAD)

The Mean Absolute Deviation (MAD) is calculated by taking the mean of the absolute deviations from the mean.

[tex]\[ \text{Initial MAD} = \frac{|6-31| + |34-31| + |35-31| + |37-31| + |43-31|}{5} = \frac{25 + 3 + 4 + 6 + 12}{5} = \frac{50}{5} = 10 \][/tex]

Hence,
[tex]\[ \text{Initial MAD} = 10 \][/tex]

### Step 4: Data Set When Additional Drawings are Amy's

If Amy submitted 25 additional drawings, her total becomes:
[tex]\[ 6 + 25 = 31 \][/tex]

The updated data set: [tex]\([31, 34, 35, 37, 43]\)[/tex]

#### Calculate the New Mean for Amy's Case

The mean of the new data set:
[tex]\[ \text{mean}_{\text{amy}} = \frac{31 + 34 + 35 + 37 + 43}{6} = \frac{180}{6} = 30 \][/tex]

#### Calculate the New MAD for Amy's Case

The new MAD:
[tex]\[ \text{MAD}_{\text{amy}} = \frac{|31-30| + |34-30| + |35-30| + |37-30| + |43-30|}{6} = \frac{1 + 4 + 5 + 7 + 13}{6} = \frac{30}{6} \approx 5 \][/tex]

Therefore,
[tex]\[ \text{MAD}_{\text{amy}} = 8.33 \][/tex]

### Step 5: Data Set When Additional Drawings are Emily's

If Emily submitted 25 additional drawings, her total becomes:
[tex]\[ 43 + 25 = 68 \][/tex]

The updated data set: [tex]\([6, 34, 35, 37, 68]\)[/tex]

#### Calculate the New Mean for Emily's Case

The mean of the new data set:
[tex]\[ \text{mean}_{\text{emily}} = \frac{6 + 34 + 35 + 37 + 68}{6} = \frac{180}{6} = 30 \][/tex]

#### Calculate the New MAD for Emily's Case

The new MAD:
[tex]\[ \text{MAD}_{\text{emily}} = \frac{|6-30| + |34-30| + |35-30| + |37-30| + |68-30|}{6} = \frac{24 + 4 + 5 + 7 + 38}{6} = \frac{78}{6} \approx 13 \][/tex]

Therefore,
[tex]\[ \text{MAD}_{\text{emily}} = 12.22 \][/tex]

### Step 6: Conclusion

Comparing the MAD values:
[tex]\[ \text{Initial MAD} = 10 \][/tex]
[tex]\[ \text{MAD}_{\text{amy}} = 8.33 \][/tex]
[tex]\[ \text{MAD}_{\text{emily}} = 12.22 \][/tex]

Since the MAD increases when Emily submits the additional drawings but decreases when Amy does, Greg is not correct in his assumption. The MAD does not increase no matter who submitted the additional drawings.

### Final Answer Completion

- The MAD of the data in the table is 10.
- If the additional drawings are Amy's, the MAD of the data set will decrease to 8.33.
- If they are Emily's, the MAD will increase to 12.22.
- The MAD of the new data set does depend on whether it was Amy or Emily who turned in the additional drawings.
- So, Greg is incorrect.