Ask a New Question
Menu

Use the data set to determine which statements are correct. Check the four that apply.

23, 45, 39, 57, 31, 53, 28, 50, 42, 36(4 points)
Responses

The upper quartile is 53.
The upper quartile is 53.

The upper quartile is 50.
The upper quartile is 50.

The median is 41.
The median is 41.

The lower quartile is 36.
The lower quartile is 36.

The median is 40.5.
The median is 40.5.

The interquartile range is 19.
The interquartile range is 19.

The lower quartile is 31.

1 answer

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

  • 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.

  • 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:

  1. The upper quartile is 53. (Incorrect, it is 50.)
  2. The upper quartile is 50. (Correct.)
  3. The median is 41. (Incorrect, it is 40.5.)
  4. The lower quartile is 36. (Incorrect, it is 31.)
  5. The median is 40.5. (Correct.)
  6. The interquartile range is 19. (Correct.)
  7. 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.