Answer :
To determine how to group the class values based on accuracy relative to aluminum's true density of [tex]\(2.7 \, \text{g/ml}\)[/tex], we can start by analyzing each recorded density value:
- [tex]\(1^{\text{st}}\)[/tex] hour: [tex]\(3.1 \, \text{g/ml}\)[/tex]
- [tex]\(2^{\text{nd}}\)[/tex] hour: [tex]\(3.05 \, \text{g/ml}\)[/tex]
- [tex]\(3^{\text{rd}}\)[/tex] hour: [tex]\(1.9 \, \text{g/ml}\)[/tex]
- [tex]\(4^{\text{th}}\)[/tex] hour: [tex]\(235 \, \text{g/ml}\)[/tex]
- [tex]\(5^{\text{th}}\)[/tex] hour: [tex]\(4.2 \, \text{g/ml}\)[/tex]
- [tex]\(6^{\text{th}}\)[/tex] hour: [tex]\(4.0 \, \text{g/ml}\)[/tex]
The true density of aluminum is [tex]\(2.7 \, \text{g/ml}\)[/tex].
Next, we will classify the values into two groups:
1. Values under the true density ([tex]\(2.7 \, \text{g/ml}\)[/tex]).
2. Values over the true density ([tex]\(2.7 \, \text{g/ml}\)[/tex]).
Sorting the density values, we get the following classifications:
Group 1: Values under the true density
- [tex]\(3^{\text{rd}}\)[/tex] hour: [tex]\(1.9 \, \text{g/ml}\)[/tex]
Group 2: Values over the true density
- [tex]\(1^{\text{st}}\)[/tex] hour: [tex]\(3.1 \, \text{g/ml}\)[/tex]
- [tex]\(2^{\text{nd}}\)[/tex] hour: [tex]\(3.05 \, \text{g/ml}\)[/tex]
- [tex]\(4^{\text{th}}\)[/tex] hour: [tex]\(235 \, \text{g/ml}\)[/tex]
- [tex]\(5^{\text{th}}\)[/tex] hour: [tex]\(4.2 \, \text{g/ml}\)[/tex]
- [tex]\(6^{\text{th}}\)[/tex] hour: [tex]\(4.0 \, \text{g/ml}\)[/tex]
Now, we compare these groups with the provided answer options:
Option a:
- Group: [tex]\(1^{\text{st}}, 3^{\text{rd}}, 5^{\text{th}},\)[/tex] and [tex]\(6^{\text{th}}\)[/tex] hours, with one recorded decimal place in their values
- Group: [tex]\(2^{\text{nd}}\)[/tex] and [tex]\(4^{\text{th}}\)[/tex] hours, with two recorded decimal places in their values
Option b:
- Group: [tex]\(3^{\text{rd}}\)[/tex] and [tex]\(4^{\text{th}}\)[/tex] hours, with values under the true density
- Group: [tex]\(1^{\text{st}}, 2^{\text{nd}}, 5^{\text{th}},\)[/tex] and [tex]\(6^{\text{th}}\)[/tex] hours, with values over the true density
Option c:
- Group: [tex]\(3^{\text{rd}}\)[/tex] hour, with a 1 value
- Group [tex]\(4^{\text{th}}\)[/tex] hour, with a 2 value
- Group [tex]\(1^{\text{st}}\)[/tex] and [tex]\(2^{\text{nd}}\)[/tex] hours, with 3 values
- Group [tex]\(5^{\text{th}}\)[/tex] and [tex]\(6^{\text{th}}\)[/tex] hours, with 4 values
Option d:
- Group all classes as accurate
Based on the classifications, the correct grouping aligns with Option b:
- Group: [tex]\(3^{\text{rd}}\)[/tex] hour ([tex]\(1.9 \, \text{g/ml}\)[/tex]) with a value under the true density.
- Group: [tex]\(1^{\text{st}}\)[/tex] hour ([tex]\(3.1 \, \text{g/ml}\)[/tex]), [tex]\(2^{\text{nd}}\)[/tex] hour ([tex]\(3.05 \, \text{g/ml}\)[/tex]), [tex]\(4^{\text{th}}\)[/tex] hour ([tex]\(235 \, \text{g/ml}\)[/tex]), [tex]\(5^{\text{th}}\)[/tex] hour ([tex]\(4.2 \, \text{g/ml}\)[/tex]), and [tex]\(6^{\text{th}}\)[/tex] hour ([tex]\(4.0 \, \text{g/ml}\)[/tex]) with values over the true density.
Final Answer:
b. Group: [tex]\(3^{\text{rd}}\)[/tex] and [tex]\(4^{\text{th}}\)[/tex] hours, with values under the true density
Group: [tex]\(1^{\text{st}}, 2^{\text{nd}}, 5^{\text{th}},\)[/tex] and [tex]\(6^{\text{th}}\)[/tex] hours, with values over the true density
- [tex]\(1^{\text{st}}\)[/tex] hour: [tex]\(3.1 \, \text{g/ml}\)[/tex]
- [tex]\(2^{\text{nd}}\)[/tex] hour: [tex]\(3.05 \, \text{g/ml}\)[/tex]
- [tex]\(3^{\text{rd}}\)[/tex] hour: [tex]\(1.9 \, \text{g/ml}\)[/tex]
- [tex]\(4^{\text{th}}\)[/tex] hour: [tex]\(235 \, \text{g/ml}\)[/tex]
- [tex]\(5^{\text{th}}\)[/tex] hour: [tex]\(4.2 \, \text{g/ml}\)[/tex]
- [tex]\(6^{\text{th}}\)[/tex] hour: [tex]\(4.0 \, \text{g/ml}\)[/tex]
The true density of aluminum is [tex]\(2.7 \, \text{g/ml}\)[/tex].
Next, we will classify the values into two groups:
1. Values under the true density ([tex]\(2.7 \, \text{g/ml}\)[/tex]).
2. Values over the true density ([tex]\(2.7 \, \text{g/ml}\)[/tex]).
Sorting the density values, we get the following classifications:
Group 1: Values under the true density
- [tex]\(3^{\text{rd}}\)[/tex] hour: [tex]\(1.9 \, \text{g/ml}\)[/tex]
Group 2: Values over the true density
- [tex]\(1^{\text{st}}\)[/tex] hour: [tex]\(3.1 \, \text{g/ml}\)[/tex]
- [tex]\(2^{\text{nd}}\)[/tex] hour: [tex]\(3.05 \, \text{g/ml}\)[/tex]
- [tex]\(4^{\text{th}}\)[/tex] hour: [tex]\(235 \, \text{g/ml}\)[/tex]
- [tex]\(5^{\text{th}}\)[/tex] hour: [tex]\(4.2 \, \text{g/ml}\)[/tex]
- [tex]\(6^{\text{th}}\)[/tex] hour: [tex]\(4.0 \, \text{g/ml}\)[/tex]
Now, we compare these groups with the provided answer options:
Option a:
- Group: [tex]\(1^{\text{st}}, 3^{\text{rd}}, 5^{\text{th}},\)[/tex] and [tex]\(6^{\text{th}}\)[/tex] hours, with one recorded decimal place in their values
- Group: [tex]\(2^{\text{nd}}\)[/tex] and [tex]\(4^{\text{th}}\)[/tex] hours, with two recorded decimal places in their values
Option b:
- Group: [tex]\(3^{\text{rd}}\)[/tex] and [tex]\(4^{\text{th}}\)[/tex] hours, with values under the true density
- Group: [tex]\(1^{\text{st}}, 2^{\text{nd}}, 5^{\text{th}},\)[/tex] and [tex]\(6^{\text{th}}\)[/tex] hours, with values over the true density
Option c:
- Group: [tex]\(3^{\text{rd}}\)[/tex] hour, with a 1 value
- Group [tex]\(4^{\text{th}}\)[/tex] hour, with a 2 value
- Group [tex]\(1^{\text{st}}\)[/tex] and [tex]\(2^{\text{nd}}\)[/tex] hours, with 3 values
- Group [tex]\(5^{\text{th}}\)[/tex] and [tex]\(6^{\text{th}}\)[/tex] hours, with 4 values
Option d:
- Group all classes as accurate
Based on the classifications, the correct grouping aligns with Option b:
- Group: [tex]\(3^{\text{rd}}\)[/tex] hour ([tex]\(1.9 \, \text{g/ml}\)[/tex]) with a value under the true density.
- Group: [tex]\(1^{\text{st}}\)[/tex] hour ([tex]\(3.1 \, \text{g/ml}\)[/tex]), [tex]\(2^{\text{nd}}\)[/tex] hour ([tex]\(3.05 \, \text{g/ml}\)[/tex]), [tex]\(4^{\text{th}}\)[/tex] hour ([tex]\(235 \, \text{g/ml}\)[/tex]), [tex]\(5^{\text{th}}\)[/tex] hour ([tex]\(4.2 \, \text{g/ml}\)[/tex]), and [tex]\(6^{\text{th}}\)[/tex] hour ([tex]\(4.0 \, \text{g/ml}\)[/tex]) with values over the true density.
Final Answer:
b. Group: [tex]\(3^{\text{rd}}\)[/tex] and [tex]\(4^{\text{th}}\)[/tex] hours, with values under the true density
Group: [tex]\(1^{\text{st}}, 2^{\text{nd}}, 5^{\text{th}},\)[/tex] and [tex]\(6^{\text{th}}\)[/tex] hours, with values over the true density
