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Sagot :
Sure, let's calculate the coefficient of correlation for the given data step-by-step.
### Step 1: Organize the Data
We have two sets of data:
- Income (RS.): 100, 200, 300, 400, 500, 600
- Weight (Ib): 120, 130, 140, 150, 160, 170
### Step 2: Calculate the Mean of Each Data Set
The mean is calculated by summing up all the values and then dividing by the number of values.
[tex]\[ \text{Mean of Income (RS.)} = \frac{100 + 200 + 300 + 400 + 500 + 600}{6} = \frac{2100}{6} = 350.0 \][/tex]
[tex]\[ \text{Mean of Weight (Ib)} = \frac{120 + 130 + 140 + 150 + 160 + 170}{6} = \frac{870}{6} = 145.0 \][/tex]
### Step 3: Calculate the Deviations from the Mean
Next, we calculate the deviation of each value from the mean.
- Deviations of Income:
- [tex]\(100 - 350 = -250\)[/tex]
- [tex]\(200 - 350 = -150\)[/tex]
- [tex]\(300 - 350 = -50\)[/tex]
- [tex]\(400 - 350 = 50\)[/tex]
- [tex]\(500 - 350 = 150\)[/tex]
- [tex]\(600 - 350 = 250\)[/tex]
- Deviations of Weight:
- [tex]\(120 - 145 = -25\)[/tex]
- [tex]\(130 - 145 = -15\)[/tex]
- [tex]\(140 - 145 = -5\)[/tex]
- [tex]\(150 - 145 = 5\)[/tex]
- [tex]\(160 - 145 = 15\)[/tex]
- [tex]\(170 - 145 = 25\)[/tex]
### Step 4: Calculate the Covariance
The covariance is calculated as the average of the product of the deviations of each pair of values.
[tex]\[ \text{Covariance} = \frac{(-250 \cdot -25) + (-150 \cdot -15) + (-50 \cdot -5) + (50 \cdot 5) + (150 \cdot 15) + (250 \cdot 25)}{6} \][/tex]
[tex]\[ = \frac{6250 + 2250 + 250 + 250 + 2250 + 6250}{6} = \frac{17000}{6} = 2833.33 \][/tex]
### Step 5: Calculate the Standard Deviation of Each Data Set
The standard deviation measures the amount of variation in the data. It is calculated as the square root of the variance.
- Standard Deviation of Income:
[tex]\[ \text{Variance of Income} = \frac{((-250)^2 + (-150)^2 + (-50)^2 + 50^2 + 150^2 + 250^2)}{6} \][/tex]
[tex]\[ = \frac{62500 + 22500 + 2500 + 2500 + 22500 + 62500}{6} = 29166.67 \][/tex]
[tex]\[ \text{Standard Deviation of Income} = \sqrt{29166.67} = 170.78 \][/tex]
- Standard Deviation of Weight:
[tex]\[ \text{Variance of Weight} = \frac{((-25)^2 + (-15)^2 + (-5)^2 + 5^2 + 15^2 + 25^2)}{6} \][/tex]
[tex]\[ = \frac{625 + 225 + 25 + 25 + 225 + 625}{6} = 2250 \][/tex]
[tex]\[ \text{Standard Deviation of Weight} = \sqrt{2250} = 47.43 \][/tex]
### Step 6: Calculate the Correlation Coefficient
The correlation coefficient is given by:
[tex]\[ r = \frac{\text{Covariance}}{(\text{Standard Deviation of Income} \cdot \text{Standard Deviation of Weight})} \][/tex]
[tex]\[ r = \frac{2833.33}{170.78 \cdot 47.43} = 0.35 \][/tex]
### Conclusion
Thus, the coefficient of correlation between income and weight is approximately 0.35, indicating a moderate positive relationship between the two variables.
### Step 1: Organize the Data
We have two sets of data:
- Income (RS.): 100, 200, 300, 400, 500, 600
- Weight (Ib): 120, 130, 140, 150, 160, 170
### Step 2: Calculate the Mean of Each Data Set
The mean is calculated by summing up all the values and then dividing by the number of values.
[tex]\[ \text{Mean of Income (RS.)} = \frac{100 + 200 + 300 + 400 + 500 + 600}{6} = \frac{2100}{6} = 350.0 \][/tex]
[tex]\[ \text{Mean of Weight (Ib)} = \frac{120 + 130 + 140 + 150 + 160 + 170}{6} = \frac{870}{6} = 145.0 \][/tex]
### Step 3: Calculate the Deviations from the Mean
Next, we calculate the deviation of each value from the mean.
- Deviations of Income:
- [tex]\(100 - 350 = -250\)[/tex]
- [tex]\(200 - 350 = -150\)[/tex]
- [tex]\(300 - 350 = -50\)[/tex]
- [tex]\(400 - 350 = 50\)[/tex]
- [tex]\(500 - 350 = 150\)[/tex]
- [tex]\(600 - 350 = 250\)[/tex]
- Deviations of Weight:
- [tex]\(120 - 145 = -25\)[/tex]
- [tex]\(130 - 145 = -15\)[/tex]
- [tex]\(140 - 145 = -5\)[/tex]
- [tex]\(150 - 145 = 5\)[/tex]
- [tex]\(160 - 145 = 15\)[/tex]
- [tex]\(170 - 145 = 25\)[/tex]
### Step 4: Calculate the Covariance
The covariance is calculated as the average of the product of the deviations of each pair of values.
[tex]\[ \text{Covariance} = \frac{(-250 \cdot -25) + (-150 \cdot -15) + (-50 \cdot -5) + (50 \cdot 5) + (150 \cdot 15) + (250 \cdot 25)}{6} \][/tex]
[tex]\[ = \frac{6250 + 2250 + 250 + 250 + 2250 + 6250}{6} = \frac{17000}{6} = 2833.33 \][/tex]
### Step 5: Calculate the Standard Deviation of Each Data Set
The standard deviation measures the amount of variation in the data. It is calculated as the square root of the variance.
- Standard Deviation of Income:
[tex]\[ \text{Variance of Income} = \frac{((-250)^2 + (-150)^2 + (-50)^2 + 50^2 + 150^2 + 250^2)}{6} \][/tex]
[tex]\[ = \frac{62500 + 22500 + 2500 + 2500 + 22500 + 62500}{6} = 29166.67 \][/tex]
[tex]\[ \text{Standard Deviation of Income} = \sqrt{29166.67} = 170.78 \][/tex]
- Standard Deviation of Weight:
[tex]\[ \text{Variance of Weight} = \frac{((-25)^2 + (-15)^2 + (-5)^2 + 5^2 + 15^2 + 25^2)}{6} \][/tex]
[tex]\[ = \frac{625 + 225 + 25 + 25 + 225 + 625}{6} = 2250 \][/tex]
[tex]\[ \text{Standard Deviation of Weight} = \sqrt{2250} = 47.43 \][/tex]
### Step 6: Calculate the Correlation Coefficient
The correlation coefficient is given by:
[tex]\[ r = \frac{\text{Covariance}}{(\text{Standard Deviation of Income} \cdot \text{Standard Deviation of Weight})} \][/tex]
[tex]\[ r = \frac{2833.33}{170.78 \cdot 47.43} = 0.35 \][/tex]
### Conclusion
Thus, the coefficient of correlation between income and weight is approximately 0.35, indicating a moderate positive relationship between the two variables.
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