To analyze the dataset \(23, 45, 39, 57, 31, 53, 28, 50, 42, 36\), we will first sort the data and then find the quartiles and the median.
Step 1: Sort the data
The sorted data is: \[ 23, 28, 31, 36, 39, 42, 45, 50, 53, 57 \]
Step 2: Determine the median
The median is the middle value. For a dataset with an even number of observations, the median is the average of the two middle numbers.
With 10 data points, the middle numbers are the 5th and 6th:
- 5th: 39
- 6th: 42
So, the median is: \[ \text{Median} = \frac{39 + 42}{2} = \frac{81}{2} = 40.5 \]
Step 3: Determine the lower and upper quartiles
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Lower Quartile (Q1): This is the median of the first half of the data (lower half). The lower half is: \[ 23, 28, 31, 36, 39 \] The median of this set (middle value) is 31.
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Upper Quartile (Q3): This is the median of the second half of the data (upper half). The upper half is: \[ 42, 45, 50, 53, 57 \] The median of this set (middle value) is 50.
Step 4: Calculate the Interquartile Range (IQR)
The interquartile range is given by: \[ \text{IQR} = Q3 - Q1 = 50 - 31 = 19 \]
Now we will check which statements are correct based on our findings:
- The upper quartile is 53. (Incorrect, it is 50.)
- The upper quartile is 50. (Correct.)
- The median is 41. (Incorrect, it is 40.5.)
- The lower quartile is 36. (Incorrect, it is 31.)
- The median is 40.5. (Correct.)
- The interquartile range is 19. (Correct.)
- The lower quartile is 31. (Correct.)
Conclusion
The correct statements are:
- The upper quartile is 50.
- The median is 40.5.
- The interquartile range is 19.
- The lower quartile is 31.
So, the four correct responses from your options are:
- The upper quartile is 50.
- The median is 40.5.
- The interquartile range is 19.
- The lower quartile is 31.