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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.
Thank you for reading the article In recent years homeowners in Spain refinanced their home mortgages due to low interest rates Santana Sosa is a mortgage officer at Open Bank S. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
