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Following are the number of grams of carbohydrates in 12-ounce espresso beverages offered at a coffee shop.

45
29
13
61
18
33
21
40
41
26
27
12
30
14
19
34
22
25
42
11
47
57
32
32

What is the first and 3rd quartiles of this data

5 answers

To find the first (Q1) and third (Q3) quartiles of the given data, we first need to arrange the data in ascending order and then find the quartiles.

Step 1: Arrange the data in ascending order

The provided data points are:

45, 29, 13, 61, 18, 33, 21, 40, 41, 26, 27, 12, 30, 14, 19, 34, 22, 25, 42, 11, 47, 57, 32, 32

Sorted in ascending order:

11, 12, 13, 14, 18, 19, 21, 22, 25, 26, 27, 29, 30, 32, 32, 33, 34, 40, 41, 42, 45, 47, 57, 61

Step 2: Find the first (Q1) and third (Q3) quartiles

Total number of observations (n): 24

  • First Quartile (Q1): The median of the first half of the data. Since there are 24 data points, the first half consists of the first 12 points.

First half of data:

11, 12, 13, 14, 18, 19, 21, 22, 25, 26, 27, 29

The median of this half (Q1) is the average of the 6th and 7th numbers:

Q1 = (19 + 21) / 2 = 20
  • Third Quartile (Q3): The median of the second half of the data. The second half consists of the last 12 points.

Second half of data:

30, 32, 32, 33, 34, 40, 41, 42, 45, 47, 57, 61

The median of this half (Q3) is the average of the 6th and 7th numbers:

Q3 = (40 + 41) / 2 = 40.5

Step 3: Summary

  • First Quartile (Q1): 20
  • Third Quartile (Q3): 40.5
(b) Find the median of these data.

To find the median of the given data, we first need to ensure it's sorted, and then we can determine the median based on the total number of observations.

Step 1: Confirm the sorted data

The sorted data points are:

11, 12, 13, 14, 18, 19, 21, 22, 25, 26, 27, 29, 30, 32, 32, 33, 34, 40, 41, 42, 45, 47, 57, 61

Step 2: Find the median

Total number of observations (n): 24 (even number)

When \( n \) is even, the median is calculated as the average of the two middle numbers. The middle numbers can be found at positions \( \frac{n}{2} \) and \( \frac{n}{2} + 1 \).

  • The two middle positions in this case are:
    • \( \frac{24}{2} = 12 \) (12th position)
    • \( \frac{24}{2} + 1 = 13 \) (13th position)

The values at these positions in the sorted data are:

  • 12th value: 29
  • 13th value: 30

Step 3: Calculate the median

\[ \text{Median} = \frac{29 + 30}{2} = \frac{59}{2} = 29.5 \]

Conclusion

The median of the given data is 29.5.

(c) Find the lower and upper outlier boundaries.

To find the lower and upper outlier boundaries, we first need to calculate the interquartile range (IQR) and then determine the boundaries based on it.

Step 1: Calculate the Interquartile Range (IQR)

The Interquartile Range (IQR) is defined as follows: \[ \text{IQR} = Q3 - Q1 \]

From previous calculations:

  • First Quartile (Q1): 20
  • Third Quartile (Q3): 40.5

Calculating the IQR: \[ \text{IQR} = 40.5 - 20 = 20.5 \]

Step 2: Calculate the Lower and Upper Outlier Boundaries

The lower and upper boundaries for outliers are calculated as follows:

  • Lower boundary: \( Q1 - 1.5 \times \text{IQR} \)
  • Upper boundary: \( Q3 + 1.5 \times \text{IQR} \)

Calculating the lower boundary: \[ \text{Lower boundary} = 20 - 1.5 \times 20.5 = 20 - 30.75 = -10.75 \]

Calculating the upper boundary: \[ \text{Upper boundary} = 40.5 + 1.5 \times 20.5 = 40.5 + 30.75 = 71.25 \]

Conclusion

  • Lower outlier boundary: \(-10.75\)
  • Upper outlier boundary: \(71.25\)

Any values below \(-10.75\) or above \(71.25\) would be considered outliers in this dataset.