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Solve the given problem. Show solutions if solved manually or show the Excel computation results. PROBLEM : The amount spent on medical expenses per day is correlated with other health factors for 5 adult males. A study came up with the following table:                
Medical        amount spent
cost                   for alcohol                      weight                   Age
2100                     200                                     185                      50
2378                     250                                    200                       42
1657                    100                                     175                       37
2584                    200                                     225                       54
2658                    250                                     220                        32



1. Which of the factors has a significant effect on medical expenses? 2. Create the regression equation for the problem.​

Sagot :

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

Learn more about regression coefficient here:

https://brainly.com/question/11204559

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