The accompanying data represent the wait time​ (in minutes) for a random sample of forty visitors to an amusement park ride. Complete parts​ (a) and​ (b).
6 15 3 6 4 30 6 6 22 8
8 46 10 10 33 0 26 5 10 6
20 8 22 9 29 17 53 3 10 32
0 4 5 12 5 8 7 16 8 15
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Part 1
​(a) Determine and interpret the quartiles.


To find and interpret the quartiles, we'll organize the data and use the formulas for \( Q_1 \) (first quartile), \( Q_2 \) (second quartile or median), and \( Q_3 \) (third quartile). The quartiles divide the dataset into four equal parts.

### Given Data:
6, 15, 3, 6, 4, 30, 6, 6, 22, 8  
8, 46, 10, 10, 33, 0, 26, 5, 10, 6  
20, 8, 22, 9, 29, 17, 53, 3, 10, 32  
0, 4, 5, 12, 5, 8, 7, 16, 8, 15  

#### Step 1: Organize the data in ascending order:
0, 0, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 12, 15, 15, 16, 17, 20, 22, 22, 26, 29, 30, 32, 33, 46, 53

#### Step 2: Calculate Quartiles
- **\( Q_1 \)**: The first quartile (25th percentile) is the median of the lower half of the data.
- **\( Q_2 \)**: The second quartile is the median (50th percentile).
- **\( Q_3 \)**: The third quartile (75th percentile) is the median of the upper half of the data.

The formula for quartiles:
\[ Q_k = \frac{k(n+1)}{4} \]
where \( k = 1, 2, 3 \) and \( n = 40 \) (the number of data points).

- **\( Q_1 \):** The position of the first quartile is:
\[
Q_1 = \frac{1(40+1)}{4} = \frac{41}{4} = 10.25
\]
So, \( Q_1 \) is the 10th data point (since 0.25 indicates a small interpolation).

Looking at the ordered data, the 10th point is 6, so \( Q_1 = 6 \).

- **\( Q_2 \):** The median \( Q_2 \) (middle of the dataset) is:
\[
Q_2 = \frac{2(40+1)}{4} = \frac{82}{4} = 20.5
\]
So, \( Q_2 \) lies between the 20th and 21st data points (which are 8 and 9).

Taking the average:
\[
Q_2 = \frac{8 + 9}{2} = 8.5
\]

- **\( Q_3 \):** The position of the third quartile is:
\[
Q_3 = \frac{3(40+1)}{4} = \frac{123}{4} = 30.75
\]
So, \( Q_3 \) lies between the 30th and 31st data points (17 and 20).

Taking the average:
\[
Q_3 = \frac{17 + 20}{2} = 18.5
\]

#### Interpretation of Quartiles:
- \( Q_1 = 6 \): 25% of the visitors waited 6 minutes or less.
- \( Q_2 = 8.5 \): 50% of the visitors waited 8.5 minutes or less (median wait time).
- \( Q_3 = 18.5 \): 75% of the visitors waited 18.5 minutes or less.

Would you like more details or any questions on this?

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Here are 5 relative questions for further practice:
1. How would you calculate the interquartile range (IQR)?
2. What does it mean if a data point lies outside the quartile range?
3. How do outliers affect the interpretation of quartiles?
4. How would the quartiles change if you removed the largest value?
5. Can quartiles give insights into data symmetry or skewness?

**Tip**: Quartiles are excellent for understanding the distribution of data and identifying potential outliers.

The accompanying data represent the wait time​ (in minutes) for a random sample of forty visitors to an amusement park ride. Complete parts​ (a) and​ (b).
6 15 3 6 4 30 6 6 22 8
8 46 10 10 33 0 26 5 10 6
20 8 22 9 29 17 53 3 10 32
0 4 5 12 5 8 7 16 8 15
(a) Determine and interpret the quartiles.

Learn how to calculate and interpret quartiles using a dataset of 40 amusement park ride wait times. Understand key statistical concepts like Q1, Q2, and Q3, and their role in data distribution. This example is ideal for students in Grades 9-12.

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