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Answer :
To solve the problem of finding the median, the first quartile, and the third quartile for the refinanced loan amounts, follow these steps:
1. Sort the Data: The data is already given in ascending order.
2. Identify the Median:
- The median is the middle value of a data set when the numbers are in order. With 20 values, the median will be the average of the 10th and 11th values.
- The 10th value is 77.3 and the 11th value is 79.2.
[tex]\[
\text{Median} = \frac{77.3 + 79.2}{2} = 78.25
\][/tex]
3. Calculate the First Quartile (Q1):
- The first quartile cuts off the lowest 25% of the data. It is the value at the position [tex]\(\frac{n+1}{4}\)[/tex] where [tex]\(n\)[/tex] is the total number of observations.
- With 20 observations, this position is [tex]\(\frac{20+1}{4} = 5.25\)[/tex].
- This means the first quartile is between the 5th (61.6) and the 6th (65.5) values. To find Q1, interpolate between these values.
[tex]\[
Q1 \approx 61.6 + 0.25 \times (65.5 - 61.6) = 64.525
\][/tex]
4. Calculate the Third Quartile (Q3):
- The third quartile cuts off the highest 25% of the data. It is the value at the position [tex]\(3 \times \frac{n+1}{4}\)[/tex].
- For 20 observations, this position is [tex]\(3 \times \frac{21}{4} = 15.75\)[/tex].
- This means the third quartile is between the 15th (86.6) and the 16th (93.3) values. To find Q3, interpolate between these values.
[tex]\[
Q3 \approx 86.6 + 0.75 \times (93.3 - 86.6) = 88.275
\][/tex]
Hence, the median is 78.25, the first quartile is approximately 64.525, and the third quartile is approximately 88.275.
1. Sort the Data: The data is already given in ascending order.
2. Identify the Median:
- The median is the middle value of a data set when the numbers are in order. With 20 values, the median will be the average of the 10th and 11th values.
- The 10th value is 77.3 and the 11th value is 79.2.
[tex]\[
\text{Median} = \frac{77.3 + 79.2}{2} = 78.25
\][/tex]
3. Calculate the First Quartile (Q1):
- The first quartile cuts off the lowest 25% of the data. It is the value at the position [tex]\(\frac{n+1}{4}\)[/tex] where [tex]\(n\)[/tex] is the total number of observations.
- With 20 observations, this position is [tex]\(\frac{20+1}{4} = 5.25\)[/tex].
- This means the first quartile is between the 5th (61.6) and the 6th (65.5) values. To find Q1, interpolate between these values.
[tex]\[
Q1 \approx 61.6 + 0.25 \times (65.5 - 61.6) = 64.525
\][/tex]
4. Calculate the Third Quartile (Q3):
- The third quartile cuts off the highest 25% of the data. It is the value at the position [tex]\(3 \times \frac{n+1}{4}\)[/tex].
- For 20 observations, this position is [tex]\(3 \times \frac{21}{4} = 15.75\)[/tex].
- This means the third quartile is between the 15th (86.6) and the 16th (93.3) values. To find Q3, interpolate between these values.
[tex]\[
Q3 \approx 86.6 + 0.75 \times (93.3 - 86.6) = 88.275
\][/tex]
Hence, the median is 78.25, the first quartile is approximately 64.525, and the third quartile is approximately 88.275.
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