Question

1. what is the mean of the data? ______
2. what is the median of the data? ______
3. circle the data values that fall within the mean absolute deviation.
4. 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.

Explanation:

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 q…

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.

\(\frac{80}{16} = 5\).

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.

\(\frac{5 + 5}{2} = 5\).

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:

5

Question 14: