13. what is the mean of the data? ______
14. what is the median of the data? ______
15. circle the data values that fall within the mean absolute deviation.
16. which statements describe the distribution of the data in the box plot? select all that apply.

dot plot: x - axis labeled 1 to 10. dots: at 2 (1 dot), 3 (2 dots), 4 (3 dots), 5 (4 dots), 6 (3 dots), 7 (2 dots), 8 (1 dot).
box plot: title \pairs of pants owned\, x - axis from 0 to 20. options:
a) of the people surveyed, $\frac{1}{2}$ own 7 to 13 pairs of pants.
b) of the people surveyed, $\frac{1}{4}$ own 3 to 10 pairs of pants.
c) of the people surveyed, $\frac{1}{2}$ own 13 to 18 pairs of pants.
d) of the people surveyed, $\frac{1}{4}$ own 3 to 7 pairs of pants.
e) of the people surveyed, $\frac{3}{4}$ own 10 to 18 pairs of pants.

Question

13. what is the mean of the data? ______
14. what is the median of the data? ______
15. circle the data values that fall within the mean absolute deviation.
16. which statements describe the distribution of the data in the box plot? select all that apply.

dot plot: x - axis labeled 1 to 10. dots: at 2 (1 dot), 3 (2 dots), 4 (3 dots), 5 (4 dots), 6 (3 dots), 7 (2 dots), 8 (1 dot).
box plot: title \pairs of pants owned\, x - axis from 0 to 20. options:
a) of the people surveyed, $\frac{1}{2}$ own 7 to 13 pairs of pants.
b) of the people surveyed, $\frac{1}{4}$ own 3 to 10 pairs of pants.
c) of the people surveyed, $\frac{1}{2}$ own 13 to 18 pairs of pants.
d) of the people surveyed, $\frac{1}{4}$ own 3 to 7 pairs of pants.
e) of the people surveyed, $\frac{3}{4}$ own 10 to 18 pairs of pants.

Accepted Answer

### Question 13: Mean Calculation
First, we need to determine the frequency of each data point from the dot plot:
- At 2: 1 dot
- At 3: 2 dots
- At 4: 3 dots
- At 5: 4 dots
- At 6: 3 dots
- At 7: 2 dots
- At 8: 1 dot

Now, calculate the sum of (value × frequency) and the total number of data points.

#### Step 1: Calculate Total Sum
$$
\begin{align*}
(2×1) + (3×2) + (4×3) + (5×4) + (6×3) + (7×2) + (8×1) &= 2 + 6 + 12 + 20 + 18 + 14 + 8 \
&= 2 + 6 = 8; \, 8 + 12 = 20; \, 20 + 20 = 40; \, 40 + 18 = 58; \, 58 + 14 = 72; \, 72 + 8 = 80
\end{align*}
$$

#### Step 2: Calculate Total Number of Data Points
$$
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
$$

#### Step 3: Calculate Mean
Mean = $\frac{\text{Total Sum}}{\text{Total Number of Data Points}} = \frac{80}{16} = 5$

### Question 14: Median Calculation
The total number of data points is 16 (even). For an even number of data points, the median is the average of the $\frac{n}{2}$-th and $(\frac{n}{2} + 1)$-th values when sorted.

#### Step 1: Determine the Position of Median Values
$n = 16$, so $\frac{n}{2} = 8$ and $\frac{n}{2} + 1 = 9$. We need the 8th and 9th values in the sorted data.

#### Step 2: Find the 8th and 9th Values
Let's list the cumulative frequencies:
- Up to 2: 1
- Up to 3: 1 + 2 = 3
- Up to 4: 3 + 3 = 6
- Up to 5: 6 + 4 = 10

So the 8th and 9th values are both 5 (since cumulative frequency at 5 is 10, which includes the 8th and 9th values).

#### Step 3: Calculate Median
Median = $\frac{5 + 5}{2} = 5$

### Question 15: Mean Absolute Deviation (MAD)
First, we already know the mean ($\mu = 5$). Now, calculate the absolute deviation for each data point:

- For 2: $|2 - 5| = 3$
- For 3: $|3 - 5| = 2$ (frequency 2)
- For 4: $|4 - 5| = 1$ (frequency 3)
- For 5: $|5 - 5| = 0$ (frequency 4)
- For 6: $|6 - 5| = 1$ (frequency 3)
- For 7: $|7 - 5| = 2$ (frequency 2)
- For 8: $|8 - 5| = 3$ (frequency 1)

#### Step 1: Calculate Total Absolute Deviations
$$
\begin{align*}
(3×1) + (2×2) + (1×3) + (0×4) + (1×3) + (2×2) + (3×1) &= 3 + 4 + 3 + 0 + 3 + 4 + 3 \
&= 3 + 4 = 7; \, 7 + 3 = 10; \, 10 + 0 = 10; \, 10 + 3 = 13; \, 13 + 4 = 17; \, 17 + 3 = 20
\end{align*}
$$

#### Step 2: Calculate MAD
MAD = $\frac{\text{Total Absolute Deviations}}{\text{Total Number of Data Points}} = \frac{20}{16} = 1.25$

#### Step 3: Determine Range for MAD
The range within the mean absolute deviation is $\mu - \text{MAD}$ to $\mu + \text{MAD}$, which is $5 - 1.25 = 3.75$ to $5 + 1.25 = 6.25$.

Now, identify data points within 3.75 to 6.25:
- 4 (frequency 3)
- 5 (frequency 4)
- 6 (frequency 3)

So we circle the data values 4, 5, and 6 (the dots at positions 4, 5, 6 on the number line).

### Question 16: Box Plot Interpretation
A box plot shows:
- Minimum (leftmost dot)
- Q1 (25th percentile, left edge of box)
- Median (line inside box)
- Q3 (75th percentile, right edge of box)
- Maximum (rightmost dot)

From the box plot:
- Minimum: 3 (leftmost dot)
- Q1: 7 (left edge of box)
- Median: 10 (line inside box)
- Q3: 13 (right edge of box)
- Maximum: 18 (rightmost dot)

Now, analyze each option:

- **Option A**: $\frac{1}{2}$ own 7 to 13 pairs. The box (Q1 to Q3) represents the middle 50%? No, wait, Q1 to Q3 is the interquartile range (IQR), which is 25th to 75th percentile (50% of data). Wait, no: Q1 is 25th, Q3 is 75th, so the middle 50% is between Q1 and Q3. Wait, the box is from Q1 (7) to Q3 (13), so 50% of data is between 7 and 13. So Option A says $\frac{1}{2}$ (50%) own 7 to 13, which is correct.

- **Option B**: $\frac{1}{4}$ own 3 to 10. Q1 is 7, Median is 10. The lower quartile (25%) is from minimum (3) to Q1 (7). So 25% ( $\frac{1}{4}$) is from 3 to 7, not 3 to 10. So B is incorrect.

- **Option C**: $\frac{1}{2}$ own 13 to 18. The upper quartile (25%) is from Q3 (13) to maximum (18). So 25% ( $\frac{1}{4}$) own 13 to 18, not $\frac{1}{2}$. So C is incorrect.

- **Option D**: $\frac{1}{4}$ own 3 to 7. The lower quartile (25%) is from minimum (3) to Q1 (7). So 25% ( $\frac{1}{4}$) own 3 to 7. This is correct.

- **Option E**: $\frac{3}{4}$ own 10 to 18. The median is 10, Q3 is 13, maximum is 18. From median (10) to maximum (18) is 50% (median to max) plus? Wait, Q3 is 75th percentile, so from Q1 (7) to max (18) is 75% (25th to 100th). But 10 to 18: median (10) to max (18) is 50% (50th to 100th), plus from Q3 (13) to max (18) is 25%? No, this is confusing. Wait, the upper quartile (25%) is Q3 to max (13 to 18). The median to Q3 is 25% (50th to 75th). So from 10 to 18: median (10) to max (18) is 50% (50th to 100th). So $\frac{3}{4}$ (75%) would be from Q1 (7) to max (18), not 10 to 18. So E is incorrect.

### Final Answers

#### Question 13:
# Explanation:
## Step1: Count frequencies of each data point.
Data points: 2(1), 3(2), 4(3), 5(4), 6(3), 7(2), 8(1).
## Step2: Calculate total sum.
$2×1 + 3×2 + 4×3 + 5×4 + 6×3 + 7×2 + 8×1 = 80$.
## Step3: Calculate total number of data points.
$1 + 2 + 3 + 4 + 3 + 2 + 1 = 16$.
## Step4: Calculate mean.
$\frac{80}{16} = 5$.
# Answer: 5

#### Question 14:
# Explanation:
## Step1: Determine n = 16 (even).
Median is average of 8th and 9th values.
## Step2: Find 8th and 9th values.
Cumulative frequencies: 2(1), 3(3), 4(6), 5(10). 8th and 9th values are 5.
## Step3: Calculate median.
$\frac{5 + 5}{2} = 5$.
# Answer: 5

#### Question 15:
# Explanation:
## Step1: Calculate MAD.
Mean = 5. Absolute deviations: 3(1), 2(2), 1(3), 0(4), 1(3), 2(2), 3(1). Total deviations = 20. MAD = $\frac{20}{16} = 1.25$.
## Step2: Determine range.
$5 - 1.25 = 3.75$ to $5 + 1.25 = 6.25$.
## Step3: Identify data points.
Data points 4, 5, 6 (dots at 4, 5, 6) fall within this range.
# Answer: Circle the data values 4, 5, and 6 (dots at positions 4, 5, 6 on the number line).

#### Question 16:
# Explanation:
- **Option A**: Box (Q1 to Q3) is 7 to 13, representing 50% ( $\frac{1}{2}$) of data. Correct.
- **Option B**: Lower quartile (3 to 7) is 25%, not 3 to 10. Incorrect.
- **Option C**: Upper quartile (13 to 18) is 25%, not 50%. Incorrect.
- **Option D**: Lower quartile (3 to 7) is 25% ( $\frac{1}{4}$). Correct.
- **Option E**: 10 to 18 is not 75% of data. Incorrect.
# Answer: A. Of the people surveyed, $\frac{1}{2}$ own 7 to 13 pairs of pants, D. Of the people surveyed, $\frac{1}{4}$ own 3 to 7 pairs of pants.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 13: Mean Calculation

Step 1: Calculate Total Sum

Step 2: Calculate Total Number of Data Points

Step 3: Calculate Mean

Question 14: Median Calculation

Step 1: Determine the Position of Median Values

Step 2: Find the 8th and 9th Values

Step 3: Calculate Median

Question 15: Mean Absolute Deviation (MAD)

Step 1: Calculate Total Absolute Deviations

Step 2: Calculate MAD

Step 3: Determine Range for MAD

Question 16: Box Plot Interpretation

Step1: Count frequencies of each data point.

Step2: Calculate total sum.

Step3: Calculate total number of data points.

Step4: Calculate mean.

Step1: Determine n = 16 (even).

Step2: Find 8th and 9th values.

Step3: Calculate median.

Step1: Calculate MAD.

Step2: Determine range.

Step3: Identify data points.

Answer:

Question 14: