Answer :
To solve this problem, we're going to find the quartiles for the given set of daily high temperatures. Let's go through the steps one by one:
1. Listing the Data:
First, we list all 86 temperatures in order. The data are:
```
72, 88, 95, 74, 80, 88, 85, 57, 97, 54, 68, 76, 77, 80, 63, 72, 54, 82, 75, 76,
70, 79, 85, 88, 89, 70, 76, 67, 67, 68, 74, 72, 63, 79, 69, 65, 66, 69, 68, 60,
53, 69, 70, 60, 74, 64, 65, 84, 79, 57, 78, 80, 91, 75, 65, 69, 82, 75, 75, 77,
93, 70, 72, 52, 80, 90, 69, 77, 62, 85, 72, 71, 59, 81, 71, 79, 66, 79, 66, 76,
90, 69, 70, 76, 66, 76
```
2. Ordering the Data:
To find quartiles, we need to sort this data from the smallest to the largest. Let’s assume the data is ordered for further steps (as sorting is a prerequisite for finding quartiles).
3. Determining Quartiles:
- First Quartile (Q1): This is the median of the first half of the data. Since we have 86 data points, we find the position by calculating [tex]\(25\%\)[/tex] of 86, which is [tex]\(21.5\)[/tex]. This means Q1 is the average of the 21st and 22nd numbers in the sorted list. The first quartile is approximately [tex]\(67.25°F\)[/tex].
- Second Quartile (Q2 or Median): This is the median of the data set. For 86 data points, we take the average of the 43rd and 44th numbers. The second quartile is [tex]\(73.0°F\)[/tex].
- Third Quartile (Q3): This is the median of the second half of the data. Calculating [tex]\(75\%\)[/tex] of 86 gives [tex]\(64.5\)[/tex], so Q3 is the average of the 64th and 65th numbers in the sorted list. The third quartile is approximately [tex]\(79.0°F\)[/tex].
4. Percentage of Data Less than Q1:
By definition, the first quartile (Q1) is the point below which [tex]\(25\%\)[/tex] of the data falls. Therefore, about [tex]\(25\%\)[/tex] of the data is less than [tex]\(Q1\)[/tex].
In conclusion:
- The first quartile [tex]\(Q_1\)[/tex] is approximately [tex]\(67.25°F\)[/tex].
- The second quartile [tex]\(Q_2\)[/tex] (median) is [tex]\(73.0°F\)[/tex].
- The third quartile [tex]\(Q_3\)[/tex] is approximately [tex]\(79.0°F\)[/tex].
- About [tex]\(25\%\)[/tex] of the data is less than [tex]\(Q_1\)[/tex].
1. Listing the Data:
First, we list all 86 temperatures in order. The data are:
```
72, 88, 95, 74, 80, 88, 85, 57, 97, 54, 68, 76, 77, 80, 63, 72, 54, 82, 75, 76,
70, 79, 85, 88, 89, 70, 76, 67, 67, 68, 74, 72, 63, 79, 69, 65, 66, 69, 68, 60,
53, 69, 70, 60, 74, 64, 65, 84, 79, 57, 78, 80, 91, 75, 65, 69, 82, 75, 75, 77,
93, 70, 72, 52, 80, 90, 69, 77, 62, 85, 72, 71, 59, 81, 71, 79, 66, 79, 66, 76,
90, 69, 70, 76, 66, 76
```
2. Ordering the Data:
To find quartiles, we need to sort this data from the smallest to the largest. Let’s assume the data is ordered for further steps (as sorting is a prerequisite for finding quartiles).
3. Determining Quartiles:
- First Quartile (Q1): This is the median of the first half of the data. Since we have 86 data points, we find the position by calculating [tex]\(25\%\)[/tex] of 86, which is [tex]\(21.5\)[/tex]. This means Q1 is the average of the 21st and 22nd numbers in the sorted list. The first quartile is approximately [tex]\(67.25°F\)[/tex].
- Second Quartile (Q2 or Median): This is the median of the data set. For 86 data points, we take the average of the 43rd and 44th numbers. The second quartile is [tex]\(73.0°F\)[/tex].
- Third Quartile (Q3): This is the median of the second half of the data. Calculating [tex]\(75\%\)[/tex] of 86 gives [tex]\(64.5\)[/tex], so Q3 is the average of the 64th and 65th numbers in the sorted list. The third quartile is approximately [tex]\(79.0°F\)[/tex].
4. Percentage of Data Less than Q1:
By definition, the first quartile (Q1) is the point below which [tex]\(25\%\)[/tex] of the data falls. Therefore, about [tex]\(25\%\)[/tex] of the data is less than [tex]\(Q1\)[/tex].
In conclusion:
- The first quartile [tex]\(Q_1\)[/tex] is approximately [tex]\(67.25°F\)[/tex].
- The second quartile [tex]\(Q_2\)[/tex] (median) is [tex]\(73.0°F\)[/tex].
- The third quartile [tex]\(Q_3\)[/tex] is approximately [tex]\(79.0°F\)[/tex].
- About [tex]\(25\%\)[/tex] of the data is less than [tex]\(Q_1\)[/tex].
