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Sagot :
Sure, let's create the frequency distribution table based on the given survey data. We'll fill in the values step-by-step.
Given data:
[tex]\[ 24, 19, 21, 27, 20, 17, 17, 32, 22, 26, 18, 13, 23, 30, 6, 13, 18, 22, 34, 16, 18, 23, 15, 19, 28, 25, 25, 20, 17, 15 \][/tex]
First of all, we need to determine the classes:
[tex]\[ [6 ; 11), [11 ; 16), [16 ; 21), [21 ; 26), [26 ; 31), [31 ; 36) \][/tex]
### Step-by-Step Computation
1. Count (Number of data points in each class):
- [tex]\( [6 ; 11) \)[/tex]: 1 data point
- [tex]\( [11 ; 16) \)[/tex]: 4 data points
- [tex]\( [16 ; 21) \)[/tex]: 11 data points
- [tex]\( [21 ; 26) \)[/tex]: 8 data points
- [tex]\( [26 ; 31) \)[/tex]: 4 data points
- [tex]\( [31 ; 36) \)[/tex]: 2 data points
- Total count: 30
2. Relative Frequency [tex]\( f \)[/tex] (Count divided by total number of data points):
- [tex]\( [6 ; 11) \)[/tex]: [tex]\(\frac{1}{30} = 0.0333\)[/tex]
- [tex]\( [11 ; 16) \)[/tex]: [tex]\(\frac{4}{30} = 0.1333\)[/tex]
- [tex]\( [16 ; 21) \)[/tex]: [tex]\(\frac{11}{30} = 0.3667\)[/tex]
- [tex]\( [21 ; 26) \)[/tex]: [tex]\(\frac{8}{30} = 0.2667\)[/tex]
- [tex]\( [26 ; 31) \)[/tex]: [tex]\(\frac{4}{30} = 0.1333\)[/tex]
- [tex]\( [31 ; 36) \)[/tex]: [tex]\(\frac{2}{30} = 0.0667\)[/tex]
3. Cumulative Frequency [tex]\( F \)[/tex] (Sum of counts up to the current class):
- [tex]\( [6 ; 11) \)[/tex]: 1
- [tex]\( [11 ; 16) \)[/tex]: 1 + 4 = 5
- [tex]\( [16 ; 21) \)[/tex]: 5 + 11 = 16
- [tex]\( [21 ; 26) \)[/tex]: 16 + 8 = 24
- [tex]\( [26 ; 31) \)[/tex]: 24 + 4 = 28
- [tex]\( [31 ; 36) \)[/tex]: 28 + 2 = 30
4. Class midpoint [tex]\( X \)[/tex] (Avg of the lower bound and upper bound of the class interval):
- [tex]\( [6 ; 11) \)[/tex]: [tex]\(\frac{6+11}{2} = 8.5\)[/tex]
- [tex]\( [11 ; 16) \)[/tex]: [tex]\(\frac{11+16}{2} = 13.5\)[/tex]
- [tex]\( [16 ; 21) \)[/tex]: [tex]\(\frac{16+21}{2} = 18.5\)[/tex]
- [tex]\( [21 ; 26) \)[/tex]: [tex]\(\frac{21+26}{2} = 23.5\)[/tex]
- [tex]\( [26 ; 31) \)[/tex]: [tex]\(\frac{26+31}{2} = 28.5\)[/tex]
- [tex]\( [31 ; 36) \)[/tex]: [tex]\(\frac{31+36}{2} = 33.5\)[/tex]
### Completed table:
[tex]\[ \begin{array}{|l|l|l|l|l|} \hline \text{Classes} & \text{Count} & f & F & \text{Class midpoint} (X) \\ \hline [6 ; 11) & 1 & 0.0333 & 1 & 8.5 \\ \hline [11 ; 16) & 4 & 0.1333 & 5 & 13.5 \\ \hline [16 ; 21) & 11 & 0.3667 & 16 & 18.5 \\ \hline [21 ; 26) & 8 & 0.2667 & 24 & 23.5 \\ \hline [26 ; 31) & 4 & 0.1333 & 28 & 28.5 \\ \hline [31 ; 36) & 2 & 0.0667 & 30 & 33.5 \\ \hline \text{Total} & 30 & 1.0000 & & \\ \hline \end{array} \][/tex]
Given data:
[tex]\[ 24, 19, 21, 27, 20, 17, 17, 32, 22, 26, 18, 13, 23, 30, 6, 13, 18, 22, 34, 16, 18, 23, 15, 19, 28, 25, 25, 20, 17, 15 \][/tex]
First of all, we need to determine the classes:
[tex]\[ [6 ; 11), [11 ; 16), [16 ; 21), [21 ; 26), [26 ; 31), [31 ; 36) \][/tex]
### Step-by-Step Computation
1. Count (Number of data points in each class):
- [tex]\( [6 ; 11) \)[/tex]: 1 data point
- [tex]\( [11 ; 16) \)[/tex]: 4 data points
- [tex]\( [16 ; 21) \)[/tex]: 11 data points
- [tex]\( [21 ; 26) \)[/tex]: 8 data points
- [tex]\( [26 ; 31) \)[/tex]: 4 data points
- [tex]\( [31 ; 36) \)[/tex]: 2 data points
- Total count: 30
2. Relative Frequency [tex]\( f \)[/tex] (Count divided by total number of data points):
- [tex]\( [6 ; 11) \)[/tex]: [tex]\(\frac{1}{30} = 0.0333\)[/tex]
- [tex]\( [11 ; 16) \)[/tex]: [tex]\(\frac{4}{30} = 0.1333\)[/tex]
- [tex]\( [16 ; 21) \)[/tex]: [tex]\(\frac{11}{30} = 0.3667\)[/tex]
- [tex]\( [21 ; 26) \)[/tex]: [tex]\(\frac{8}{30} = 0.2667\)[/tex]
- [tex]\( [26 ; 31) \)[/tex]: [tex]\(\frac{4}{30} = 0.1333\)[/tex]
- [tex]\( [31 ; 36) \)[/tex]: [tex]\(\frac{2}{30} = 0.0667\)[/tex]
3. Cumulative Frequency [tex]\( F \)[/tex] (Sum of counts up to the current class):
- [tex]\( [6 ; 11) \)[/tex]: 1
- [tex]\( [11 ; 16) \)[/tex]: 1 + 4 = 5
- [tex]\( [16 ; 21) \)[/tex]: 5 + 11 = 16
- [tex]\( [21 ; 26) \)[/tex]: 16 + 8 = 24
- [tex]\( [26 ; 31) \)[/tex]: 24 + 4 = 28
- [tex]\( [31 ; 36) \)[/tex]: 28 + 2 = 30
4. Class midpoint [tex]\( X \)[/tex] (Avg of the lower bound and upper bound of the class interval):
- [tex]\( [6 ; 11) \)[/tex]: [tex]\(\frac{6+11}{2} = 8.5\)[/tex]
- [tex]\( [11 ; 16) \)[/tex]: [tex]\(\frac{11+16}{2} = 13.5\)[/tex]
- [tex]\( [16 ; 21) \)[/tex]: [tex]\(\frac{16+21}{2} = 18.5\)[/tex]
- [tex]\( [21 ; 26) \)[/tex]: [tex]\(\frac{21+26}{2} = 23.5\)[/tex]
- [tex]\( [26 ; 31) \)[/tex]: [tex]\(\frac{26+31}{2} = 28.5\)[/tex]
- [tex]\( [31 ; 36) \)[/tex]: [tex]\(\frac{31+36}{2} = 33.5\)[/tex]
### Completed table:
[tex]\[ \begin{array}{|l|l|l|l|l|} \hline \text{Classes} & \text{Count} & f & F & \text{Class midpoint} (X) \\ \hline [6 ; 11) & 1 & 0.0333 & 1 & 8.5 \\ \hline [11 ; 16) & 4 & 0.1333 & 5 & 13.5 \\ \hline [16 ; 21) & 11 & 0.3667 & 16 & 18.5 \\ \hline [21 ; 26) & 8 & 0.2667 & 24 & 23.5 \\ \hline [26 ; 31) & 4 & 0.1333 & 28 & 28.5 \\ \hline [31 ; 36) & 2 & 0.0667 & 30 & 33.5 \\ \hline \text{Total} & 30 & 1.0000 & & \\ \hline \end{array} \][/tex]
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