Question
Use the data set to determine which are true select 4. 22,45,39,57,31,53,28,50,42,36 a. Median is 41 b. Interquartile range is 19 c. Upper quartile is 53 d. Lower quartile is 36 e. Lower quartile is 31’f. Median is 40.5 g. Upper quartile is 50
Answers
GPT-4o mini
To determine which statements about the dataset are true, we need to first sort the numbers and then calculate the median, lower quartile (Q1), upper quartile (Q3), and the interquartile range (IQR).
### Given data:
22, 45, 39, 57, 31, 53, 28, 50, 42, 36
### Step 1: Sort the data
Sorted data: **22, 28, 31, 36, 39, 42, 45, 50, 53, 57**
### Step 2: Find the median
The median is the average of the middle two numbers in a sorted list of 10 numbers (since there are an even number of observations).
Middle numbers: 39 (5th) and 42 (6th).
Median = (39 + 42) / 2 = 40.5.
### Step 3: Find the lower quartile (Q1)
To find Q1, we look at the first half of the data (the lower 5 numbers: 22, 28, 31, 36, 39).
Q1 is the median of these 5 numbers:
- Lower half: 22, 28, 31, 36, 39
- Median of lower half (3rd number) = 31.
### Step 4: Find the upper quartile (Q3)
To find Q3, we look at the second half of the data (the upper 5 numbers: 42, 45, 50, 53, 57).
Q3 is the median of these 5 numbers:
- Upper half: 42, 45, 50, 53, 57
- Median of upper half (3rd number) = 50.
### Step 5: Calculate the interquartile range (IQR)
IQR = Q3 - Q1 = 50 - 31 = 19.
### Summary of calculations:
- Median: **40.5**
- Lower quartile (Q1): **31**
- Upper quartile (Q3): **50**
- Interquartile range (IQR): **19**
### Evaluate statements:
a. Median is 41 - **False**
b. Interquartile range is 19 - **True**
c. Upper quartile is 53 - **False**
d. Lower quartile is 36 - **False**
e. Lower quartile is 31 - **True**
f. Median is 40.5 - **True**
g. Upper quartile is 50 - **True**
### True statements:
- **b. Interquartile range is 19**
- **e. Lower quartile is 31**
- **f. Median is 40.5**
- **g. Upper quartile is 50**
So, the four true selections are **b, e, f, g**.
### Given data:
22, 45, 39, 57, 31, 53, 28, 50, 42, 36
### Step 1: Sort the data
Sorted data: **22, 28, 31, 36, 39, 42, 45, 50, 53, 57**
### Step 2: Find the median
The median is the average of the middle two numbers in a sorted list of 10 numbers (since there are an even number of observations).
Middle numbers: 39 (5th) and 42 (6th).
Median = (39 + 42) / 2 = 40.5.
### Step 3: Find the lower quartile (Q1)
To find Q1, we look at the first half of the data (the lower 5 numbers: 22, 28, 31, 36, 39).
Q1 is the median of these 5 numbers:
- Lower half: 22, 28, 31, 36, 39
- Median of lower half (3rd number) = 31.
### Step 4: Find the upper quartile (Q3)
To find Q3, we look at the second half of the data (the upper 5 numbers: 42, 45, 50, 53, 57).
Q3 is the median of these 5 numbers:
- Upper half: 42, 45, 50, 53, 57
- Median of upper half (3rd number) = 50.
### Step 5: Calculate the interquartile range (IQR)
IQR = Q3 - Q1 = 50 - 31 = 19.
### Summary of calculations:
- Median: **40.5**
- Lower quartile (Q1): **31**
- Upper quartile (Q3): **50**
- Interquartile range (IQR): **19**
### Evaluate statements:
a. Median is 41 - **False**
b. Interquartile range is 19 - **True**
c. Upper quartile is 53 - **False**
d. Lower quartile is 36 - **False**
e. Lower quartile is 31 - **True**
f. Median is 40.5 - **True**
g. Upper quartile is 50 - **True**
### True statements:
- **b. Interquartile range is 19**
- **e. Lower quartile is 31**
- **f. Median is 40.5**
- **g. Upper quartile is 50**
So, the four true selections are **b, e, f, g**.