Question
Use the data set to determine which statements are correct. Check all that apply.
35, 41, 18, 75, 36, 21, 62, 29, 154, 70
The median is 36.
The median is 38.5.
There is an outlier.
The lower quartile is 29.
The lower quartile is 18.
The upper quartile is 29.
The upper quartile is 70.
The interquartile range is 41.
35, 41, 18, 75, 36, 21, 62, 29, 154, 70
The median is 36.
The median is 38.5.
There is an outlier.
The lower quartile is 29.
The lower quartile is 18.
The upper quartile is 29.
The upper quartile is 70.
The interquartile range is 41.
Answers
GPT-4o mini
To determine which statements are correct, we first need to gather statistics about the provided data set:
**Data Set**: 35, 41, 18, 75, 36, 21, 62, 29, 154, 70
### Step 1: Sort the Data
First, we sort the data in ascending order:
18, 21, 29, 35, 36, 41, 62, 70, 75, 154
### Step 2: Find the Median
The median is the middle value in a sorted list. For a data set with an even number of observations, the median is the average of the two middle numbers.
In our sorted data, the middle numbers (5th and 6th) are:
- 5th: 36
- 6th: 41
The median is:
\[
\text{Median} = \frac{36 + 41}{2} = \frac{77}{2} = 38.5
\]
### Step 3: Find the Quadriles
- **Lower Quartile (Q1)**: This is the median of the first half of the data.
- First half: 18, 21, 29, 35, 36 (the median is 29).
- **Upper Quartile (Q3)**: This is the median of the second half of the data.
- Second half: 41, 62, 70, 75, 154 (the median is 70).
### Step 4: Identify Outliers
We can identify outliers using the Interquartile Range (IQR):
\[
\text{IQR} = Q3 - Q1
\]
\[
\text{IQR} = 70 - 29 = 41
\]
Outliers are often defined as points that lie more than 1.5 times the IQR above Q3 or below Q1.
- Lower boundary: \( Q1 - 1.5 \times IQR = 29 - 61.5 = -32.5 \)
- Upper boundary: \( Q3 + 1.5 \times IQR = 70 + 61.5 = 131.5 \)
Any data point below -32.5 or above 131.5 is considered an outlier. In our data set, 154 is greater than 131.5, making it an outlier.
### Summary of each statement:
1. **The median is 36.** - **Incorrect** (Median is 38.5)
2. **The median is 38.5.** - **Correct**
3. **There is an outlier.** - **Correct** (154 is an outlier)
4. **The lower quartile is 29.** - **Correct**
5. **The lower quartile is 18.** - **Incorrect**
6. **The upper quartile is 29.** - **Incorrect** (Upper quartile is 70)
7. **The upper quartile is 70.** - **Correct**
8. **The interquartile range is 41.** - **Correct**
### Correct Statements:
- The median is 38.5.
- There is an outlier.
- The lower quartile is 29.
- The upper quartile is 70.
- The interquartile range is 41.
**Data Set**: 35, 41, 18, 75, 36, 21, 62, 29, 154, 70
### Step 1: Sort the Data
First, we sort the data in ascending order:
18, 21, 29, 35, 36, 41, 62, 70, 75, 154
### Step 2: Find the Median
The median is the middle value in a sorted list. For a data set with an even number of observations, the median is the average of the two middle numbers.
In our sorted data, the middle numbers (5th and 6th) are:
- 5th: 36
- 6th: 41
The median is:
\[
\text{Median} = \frac{36 + 41}{2} = \frac{77}{2} = 38.5
\]
### Step 3: Find the Quadriles
- **Lower Quartile (Q1)**: This is the median of the first half of the data.
- First half: 18, 21, 29, 35, 36 (the median is 29).
- **Upper Quartile (Q3)**: This is the median of the second half of the data.
- Second half: 41, 62, 70, 75, 154 (the median is 70).
### Step 4: Identify Outliers
We can identify outliers using the Interquartile Range (IQR):
\[
\text{IQR} = Q3 - Q1
\]
\[
\text{IQR} = 70 - 29 = 41
\]
Outliers are often defined as points that lie more than 1.5 times the IQR above Q3 or below Q1.
- Lower boundary: \( Q1 - 1.5 \times IQR = 29 - 61.5 = -32.5 \)
- Upper boundary: \( Q3 + 1.5 \times IQR = 70 + 61.5 = 131.5 \)
Any data point below -32.5 or above 131.5 is considered an outlier. In our data set, 154 is greater than 131.5, making it an outlier.
### Summary of each statement:
1. **The median is 36.** - **Incorrect** (Median is 38.5)
2. **The median is 38.5.** - **Correct**
3. **There is an outlier.** - **Correct** (154 is an outlier)
4. **The lower quartile is 29.** - **Correct**
5. **The lower quartile is 18.** - **Incorrect**
6. **The upper quartile is 29.** - **Incorrect** (Upper quartile is 70)
7. **The upper quartile is 70.** - **Correct**
8. **The interquartile range is 41.** - **Correct**
### Correct Statements:
- The median is 38.5.
- There is an outlier.
- The lower quartile is 29.
- The upper quartile is 70.
- The interquartile range is 41.