Answer :
The correlation of each factor to the medical expenses is found by taking
the medical cost as the dependent variable.
Responses:
- The factors that have significant effect on medical expenses are alcohol cost and weight
Regression equation are;
- Alcohol cost; [tex]\underline{ \overline{y} = 5.75 \cdot \overline x + 1127.4}[/tex]
- Weight; [tex]\underline{\overline{y} = 17.892 \cdot \overline x - 1,320.9}[/tex]
- Age; [tex]\underline {\overline{y} = 4.774 \cdot \overline{x} + 2,070.1}[/tex]
Methods used to find the correlation between the factors
The given data in tabular form are presented as follows;
[tex]\begin{tabular}{|c|c|c|c|} \underline{Medical cost}&\underline{Alcohol}&\underline{Weight}&\underline{Age}\\2,100&200&185&50\\2,378&250&200&42\\1,657&100&175&37\\2,584&200&225&54\\2,658&250&220&32\end{array}\right][/tex]
The regression equation is; [tex]\overline y[/tex] = a + b·[tex]\mathbf{\overline x}[/tex]
Where;
[tex]b = \mathbf{\dfrac{\sum \left(x_i - \bar x\right) \times \left(y_i - \bar y\right) }{\sum \left(x_i - \bar x\right )^2 }}[/tex]
[tex]Regression \ coefficient, \ r = \mathbf{\dfrac{n \cdot \sum XY - \left (\sum X \right )\left (\sum Y \right )}{\sqrt{n \cdot \sum X^{2} - \left (\sum X \right )^{2}\times n \cdot \sum Y^{2} - \left (\sum Y \right )^{2}}}}[/tex]
The regression equation for each health factor are calculated as follows:
For alcohol;
[tex]b = \dfrac{86,100}{15000} = \mathbf{ 5.74}[/tex]
[tex]\overline x[/tex] = 200
[tex]\overline y[/tex] = 2275.4
a = 2275.4 - 5.74 × 200 = 1127.4
- The equation is [tex]\underline{\overline y = 1127.4 + 5.75 \cdot \overline{x}}[/tex]
The regression coefficient, where n = 5 is therefore;
[tex]r = \dfrac{5 \times 2,361,500 - 1000 \times 11377}{\sqrt{(5 \times 215000 - 1000^2) \times (5 \times 26552553 - 11377^2) } } \approx \mathbf{ 0.862}[/tex]
Weight:
[tex]b = \dfrac{33,458}{1870} \approx \mathbf{ 17.892}[/tex]
[tex]\overline x[/tex] = 201
[tex]\overline y[/tex] = 2275.4
a = 2275.4 - 17.892 × 201 =-1320.9
- The equation is [tex]\underline{\overline{y} = 17.892 \cdot \overline{x} - 1,320.9}[/tex]
The regression coefficient, is therefore;
[tex]r = \dfrac{5 \times 2,320,235- 1005 \times 11377}{\sqrt{(5 \times 203875- 1005^2) \times (5 \times 26552553 - 11377^2) } } \approx \mathbf{0.949}[/tex]
Age:
[tex]b = \dfrac{1566}{328} \approx \mathbf{4.774}[/tex]
[tex]\overline x[/tex] = 43
[tex]\overline y[/tex] = 2275.4
a = 2275.4 - 4.774 × 43 ≈ 2070.1
The equation is therefore;
- [tex]\underline{ \overline {y} = 4.774 \cdot \overline{x} + 2,070.1}[/tex]
The regression coefficient is therefore;
[tex]r = \dfrac{5 \times 490777 - 215 \times 11377}{\sqrt{(5 \times 9573 - 215^2) \times (5 \times 26552553 - 11377^2) } } \approx \mathbf{ 0.106}[/tex]
- Weight and alcohol have the highest and second highest regression coefficient and therefore, have significant effect on medical expense
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