To find the correlation coefficient (r-value) for the given data, we will first compute the necessary components, such as the means of the age and height, as well as the individual products for the correlation formula.
Given the data:
| Age (years) | Height (inches) | |-------------|-----------------| | 1 | 29 | | 2 | 34 | | 3 | 37 | | 4 | 40 | | 5 | 42 | | 6 | 44 | | 7 | 47 | | 8 | 52 | | 9 | 55 | | 10 | 60 |
Step 1: Calculate the means of age and height.
\[ \text{Mean of Age} = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10}{10} = \frac{55}{10} = 5.5 \]
\[ \text{Mean of Height} = \frac{29 + 34 + 37 + 40 + 42 + 44 + 47 + 52 + 55 + 60}{10} = \frac{444}{10} = 44.4 \]
Step 2: Calculate the components needed for the correlation formula.
- \( \sum (x - \bar{x})(y - \bar{y}) \)
- \( \sum (x - \bar{x})^2 \)
- \( \sum (y - \bar{y})^2 \)
Where:
- \( x = \text{Age} \)
- \( y = \text{Height} \)
Step 3: Calculate \( \sum (x - \bar{x})(y - \bar{y}) \):
\[ \text{For each data point:} \]
\[ 1: (1 - 5.5)(29 - 44.4) = -4.5 \times -15.4 = 69.3 \] \[ 2: (2 - 5.5)(34 - 44.4) = -3.5 \times -10.4 = 36.4 \] \[ 3: (3 - 5.5)(37 - 44.4) = -2.5 \times -7.4 = 18.5 \] \[ 4: (4 - 5.5)(40 - 44.4) = -1.5 \times -4.4 = 6.6 \] \[ 5: (5 - 5.5)(42 - 44.4) = -0.5 \times -2.4 = 1.2 \] \[ 6: (6 - 5.5)(44 - 44.4) = 0.5 \times -0.4 = -0.2 \] \[ 7: (7 - 5.5)(47 - 44.4) = 1.5 \times 2.6 = 3.9 \] \[ 8: (8 - 5.5)(52 - 44.4) = 2.5 \times 7.6 = 19 \] \[ 9: (9 - 5.5)(55 - 44.4) = 3.5 \times 10.6 = 37.4 \] \[ 10: (10 - 5.5)(60 - 44.4) = 4.5 \times 15.6 = 70.2 \]
Calculating \( \sum (x - \bar{x})(y - \bar{y}) \): \[ 69.3 + 36.4 + 18.5 + 6.6 + 1.2 - 0.2 + 3.9 + 19 + 37.4 + 70.2 = 232.0 \]
Step 4: Calculate \( \sum (x - \bar{x})^2 \):
\[ (1-5.5)^2 = 20.25, \quad (2-5.5)^2 = 12.25, \quad (3-5.5)^2 = 6.25, \quad (4-5.5)^2 = 2.25, \] \[ (5-5.5)^2 = 0.25, \quad (6-5.5)^2 = 0.25, \quad (7-5.5)^2 = 2.25, \quad (8-5.5)^2 = 6.25, \] \[ (9-5.5)^2 = 12.25, \quad (10-5.5)^2 = 20.25 \] Sum: \[ 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 = 88.5 \]
Step 5: Calculate \( \sum (y - \bar{y})^2 \):
\[ (29-44.4)^2 = 240.1, \quad (34-44.4)^2 = 108.16, \quad (37-44.4)^2 = 54.76, \] \[ (40-44.4)^2 = 19.36, \quad (42-44.4)^2 = 5.76, \quad (44-44.4)^2 = 0.16, \] \[ (47-44.4)^2 = 6.76, \quad (52-44.4)^2 = 57.76, \quad (55-44.4)^2 = 110.56, \quad (60-44.4)^2 = 243.76 \] Sum: \[ 240.1 + 108.16 + 54.76 + 19.36 + 5.76 + 0.16 + 6.76 + 57.76 + 110.56 + 243.76 = 846.04 \]
Step 6: Calculate correlation coefficient \( r \):
\[ r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \cdot \sum (y - \bar{y})^2}} \] \[ r = \frac{232.0}{\sqrt{88.5 \cdot 846.04}} \approx \frac{232.0}{\sqrt{74722.54}} = \frac{232.0}{273.0} \approx 0.851 \]
Part A:
The correlation coefficient \( r \) is approximately 0.851 (rounded to the nearest thousandths).
Part B:
The r-value implies that this data has a positive correlation.
Part C:
The strength of this correlation is strong.
Final answers:
- Part A: \( r \approx 0.851 \)
- Part B: positive
- Part C: strong
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