Answer :
Certainly! Let me guide you through the solutions for the given problems step-by-step.
### Given IQ Scores of the Employees
113, 88, 115, 112, 105, 99, 112, 140, 116, 124, 113, 72, 145, 113, 85, 111, 102, 100, 110, 119
#### 3.1.1 Determine the percentage of employees that are rated as having above average intelligence.
For this, according to the Lewis Terman scale, "above average intelligence" includes scores of 110 and above (Superior Intelligence, Very Superior Intelligence, and Genius).
Let’s count the number of scores that are 110 or more:
- Scores that are 110 or more: 113, 115, 112, 112, 140, 116, 124, 113, 145, 113, 111, 110, 119
Number of such scores: 13
Total number of employees: 20
Percentage of employees with above-average intelligence:
[tex]\[ \text{Percentage} = \left( \frac{\text{Number of above average scores}}{\text{Total number of scores}} \right) \times 100 = \left( \frac{13}{20} \right) \times 100 = 65\% \][/tex]
#### 3.1.2 Determine the mean (average) score.
The mean or average score is calculated by summing all the scores and then dividing by the number of scores.
Sum of the scores: [tex]\(113 + 88 + 115 + 112 + 105 + 99 + 112 + 140 + 116 + 124 + 113 + 72 + 145 + 113 + 85 + 111 + 102 + 100 + 110 + 119 = 2144\)[/tex]
Number of scores: 20
Mean score:
[tex]\[ \text{Mean} = \frac{\text{Sum of scores}}{\text{Number of scores}} = \frac{2144}{20} = 107.2 \][/tex]
#### 3.1.3 Determine the median value for the scores.
The median is the middle value in an ordered list of numbers. If the list has an even number of entries, the median is the average of the two middle numbers.
First, we sort the scores:
72, 85, 88, 99, 100, 102, 105, 110, 111, 112, 112, 113, 113, 113, 115, 116, 119, 124, 140, 145
Since there are 20 scores (an even number), the median will be the average of the 10th and 11th scores:
[tex]\[ \text{Median} = \frac{112 + 112}{2} = \frac{224}{2} = 112 \][/tex]
#### 3.1.4 Determine the modal value (mode) for the scores.
The mode is the value that appears most frequently in a data set.
From the sorted scores, we notice that the score 113 appears most frequently (four times).
Mode: 113
#### 3.1.5 Determine the range for the above list.
The range is the difference between the highest and lowest scores in the list.
Highest score: 145
Lowest score: 72
Range:
[tex]\[ \text{Range} = 145 - 72 = 73 \][/tex]
### Summary of Answers:
#### 3.1.1 Percentage of employees with above-average intelligence:
65%
#### 3.1.2 Mean (average) score:
107.2
#### 3.1.3 Median score:
112
#### 3.1.4 Mode score:
113
#### 3.1.5 Range of the scores:
73
### Given IQ Scores of the Employees
113, 88, 115, 112, 105, 99, 112, 140, 116, 124, 113, 72, 145, 113, 85, 111, 102, 100, 110, 119
#### 3.1.1 Determine the percentage of employees that are rated as having above average intelligence.
For this, according to the Lewis Terman scale, "above average intelligence" includes scores of 110 and above (Superior Intelligence, Very Superior Intelligence, and Genius).
Let’s count the number of scores that are 110 or more:
- Scores that are 110 or more: 113, 115, 112, 112, 140, 116, 124, 113, 145, 113, 111, 110, 119
Number of such scores: 13
Total number of employees: 20
Percentage of employees with above-average intelligence:
[tex]\[ \text{Percentage} = \left( \frac{\text{Number of above average scores}}{\text{Total number of scores}} \right) \times 100 = \left( \frac{13}{20} \right) \times 100 = 65\% \][/tex]
#### 3.1.2 Determine the mean (average) score.
The mean or average score is calculated by summing all the scores and then dividing by the number of scores.
Sum of the scores: [tex]\(113 + 88 + 115 + 112 + 105 + 99 + 112 + 140 + 116 + 124 + 113 + 72 + 145 + 113 + 85 + 111 + 102 + 100 + 110 + 119 = 2144\)[/tex]
Number of scores: 20
Mean score:
[tex]\[ \text{Mean} = \frac{\text{Sum of scores}}{\text{Number of scores}} = \frac{2144}{20} = 107.2 \][/tex]
#### 3.1.3 Determine the median value for the scores.
The median is the middle value in an ordered list of numbers. If the list has an even number of entries, the median is the average of the two middle numbers.
First, we sort the scores:
72, 85, 88, 99, 100, 102, 105, 110, 111, 112, 112, 113, 113, 113, 115, 116, 119, 124, 140, 145
Since there are 20 scores (an even number), the median will be the average of the 10th and 11th scores:
[tex]\[ \text{Median} = \frac{112 + 112}{2} = \frac{224}{2} = 112 \][/tex]
#### 3.1.4 Determine the modal value (mode) for the scores.
The mode is the value that appears most frequently in a data set.
From the sorted scores, we notice that the score 113 appears most frequently (four times).
Mode: 113
#### 3.1.5 Determine the range for the above list.
The range is the difference between the highest and lowest scores in the list.
Highest score: 145
Lowest score: 72
Range:
[tex]\[ \text{Range} = 145 - 72 = 73 \][/tex]
### Summary of Answers:
#### 3.1.1 Percentage of employees with above-average intelligence:
65%
#### 3.1.2 Mean (average) score:
107.2
#### 3.1.3 Median score:
112
#### 3.1.4 Mode score:
113
#### 3.1.5 Range of the scores:
73
