The data show the number of hours spent studying per day by a sample of 28 students. Use the​ box-and-whisker plot below to answer parts​ (a) through​ (c) below.
 
1
11
3
5
6
6
8
7
1
7
9
2
5
2
2
7
3
0
6
5
8
3
3
5
3
3
9
9


Let's work through the problem step by step. The data provided is a sample of 28 students and shows the number of hours spent studying per day:

**Data set:**
1, 11, 3, 5, 6, 6, 8, 7, 1, 7, 9, 2, 5, 2, 2, 7, 3, 0, 6, 5, 8, 3, 3, 5, 3, 3, 9, 9

To create a box-and-whisker plot (also called a box plot), we need to follow these steps:

### Step 1: Sort the data
First, let's sort the data from smallest to largest:

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 11

### Step 2: Find the five-number summary
The five-number summary consists of:
- **Minimum (Min):** The smallest number in the data set.
- **First Quartile (Q1):** The median of the lower half of the data.
- **Median (Q2):** The middle number of the data set.
- **Third Quartile (Q3):** The median of the upper half of the data.
- **Maximum (Max):** The largest number in the data set.

#### Minimum (Min)
The smallest number is **0**.

#### Maximum (Max)
The largest number is **11**.

#### Median (Q2)
The median is the middle value of the data. Since we have 28 data points (an even number), the median is the average of the 14th and 15th numbers:
- 14th value: **5**
- 15th value: **5**
  
The median (Q2) = \(\frac{5 + 5}{2} = 5\).

#### First Quartile (Q1)
To find Q1, we need the median of the lower half of the data (first 14 values). The median of this half is the average of the 7th and 8th numbers:
- 7th value: **3**
- 8th value: **3**

So, Q1 = \(\frac{3 + 3}{2} = 3\).

#### Third Quartile (Q3)
To find Q3, we take the median of the upper half of the data (last 14 values). The median of this half is the average of the 21st and 22nd numbers:
- 21st value: **7**
- 22nd value: **7**

So, Q3 = \(\frac{7 + 7}{2} = 7\).

### Step 3: Five-number summary
- Minimum: **0**
- Q1: **3**
- Median (Q2): **5**
- Q3: **7**
- Maximum: **11**

### Step 4: Box-and-Whisker Plot
Based on the five-number summary, the box plot would have the following features:
- The whiskers extend from the minimum value (0) to the maximum value (11).
- The box itself spans from Q1 (3) to Q3 (7).
- The median (5) is marked within the box.

Now, let's address the parts of the problem:

### Part (a): What is the range of the data set?
The **range** is the difference between the maximum and minimum values:
\[
\text{Range} = 11 - 0 = 11
\]

### Part (b): What is the interquartile range (IQR) of the data set?
The **interquartile range (IQR)** is the difference between the third and first quartiles:
\[
\text{IQR} = Q3 - Q1 = 7 - 3 = 4
\]

### Part (c): Are there any outliers in the data set?
To determine if there are any outliers, we use the IQR to find the lower and upper bounds:
\[
\text{Lower bound} = Q1 - 1.5 \times \text{IQR} = 3 - 1.5 \times 4 = 3 - 6 = -3
\]
\[
\text{Upper bound} = Q3 + 1.5 \times \text{IQR} = 7 + 1.5 \times 4 = 7 + 6 = 13
\]
Any data points outside the range \([-3, 13]\) are considered outliers. Since all the values in the data set are between 0 and 11, there are **no outliers** in the data set.

---

Would you like more details or have any questions about this analysis?

### Related Questions:
1. How do you interpret the interquartile range in this context?
2. What is the significance of outliers in a data set?
3. How would adding an outlier, such as a value of 15, affect the box plot?
4. Can a box-and-whisker plot show skewness in data? How?
5. How can the shape of a box plot give insights into the data's distribution?

### Tip:
Box-and-whisker plots are especially useful for visualizing the spread and skewness of data, as they highlight the central 50% of the data with the box and potential outliers with whiskers.

The data show the number of hours spent studying per day by a sample of 28 students. Use the​ box-and-whisker plot below to answer parts​ (a) through​ (c) below.

1, 11, 3, 5, 6, 6, 8, 7, 1, 7, 9, 2, 5, 2, 2, 7, 3, 0, 6, 5, 8, 3, 3, 5, 3, 3, 9, 9

Analyze the number of hours spent studying per day by 28 students using a box-and-whisker plot. This guide covers the five-number summary, interquartile range, and identification of outliers, providing a clear breakdown of the data's spread. Suitable for middle school students in Grades 6-8.

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