The attached who.csv dataset contains real-world data from 2008. The variables included follow.

Country: name of the country

LifeExp: average life expectancy for the country in years

InfantSurvival: proportion of those surviving to one year or more

Under5Survival: proportion of those surviving to five years or more

TBFree: proportion of the population without TB.

PropMD: proportion of the population who are MDs

PropRN: proportion of the population who are RNs

PersExp: mean personal expenditures on healthcare in US dollars at average exchange rate

GovtExp: mean government expenditures per capita on healthcare, US dollars at average exchange rate

TotExp: sum of personal and government expenditures.

Load data from CSV file

Country LifeExp InfantSurvival Under5Survival TBFree PropMD PropRN PersExp GovtExp TotExp
Afghanistan 42 0.835 0.743 0.99769 0.0002288 0.0005723 20 92 112
Albania 71 0.985 0.983 0.99974 0.0011431 0.0046144 169 3128 3297
Algeria 71 0.967 0.962 0.99944 0.0010605 0.0020914 108 5184 5292
Andorra 82 0.997 0.996 0.99983 0.0032973 0.0035000 2589 169725 172314
Angola 41 0.846 0.740 0.99656 0.0000704 0.0011462 36 1620 1656
Antigua and Barbuda 73 0.990 0.989 0.99991 0.0001429 0.0027738 503 12543 13046
##                 Country       LifeExp      InfantSurvival   Under5Survival  
##  Afghanistan        :  1   Min.   :40.00   Min.   :0.8350   Min.   :0.7310  
##  Albania            :  1   1st Qu.:61.25   1st Qu.:0.9433   1st Qu.:0.9253  
##  Algeria            :  1   Median :70.00   Median :0.9785   Median :0.9745  
##  Andorra            :  1   Mean   :67.38   Mean   :0.9624   Mean   :0.9459  
##  Angola             :  1   3rd Qu.:75.00   3rd Qu.:0.9910   3rd Qu.:0.9900  
##  Antigua and Barbuda:  1   Max.   :83.00   Max.   :0.9980   Max.   :0.9970  
##  (Other)            :184                                                    
##      TBFree           PropMD              PropRN             PersExp       
##  Min.   :0.9870   Min.   :0.0000196   Min.   :0.0000883   Min.   :   3.00  
##  1st Qu.:0.9969   1st Qu.:0.0002444   1st Qu.:0.0008455   1st Qu.:  36.25  
##  Median :0.9992   Median :0.0010474   Median :0.0027584   Median : 199.50  
##  Mean   :0.9980   Mean   :0.0017954   Mean   :0.0041336   Mean   : 742.00  
##  3rd Qu.:0.9998   3rd Qu.:0.0024584   3rd Qu.:0.0057164   3rd Qu.: 515.25  
##  Max.   :1.0000   Max.   :0.0351290   Max.   :0.0708387   Max.   :6350.00  
##                                                                            
##     GovtExp             TotExp      
##  Min.   :    10.0   Min.   :    13  
##  1st Qu.:   559.5   1st Qu.:   584  
##  Median :  5385.0   Median :  5541  
##  Mean   : 40953.5   Mean   : 41696  
##  3rd Qu.: 25680.2   3rd Qu.: 26331  
##  Max.   :476420.0   Max.   :482750  
## 

1. Provide a scatterplot of LifeExp~TotExp, and run simple linear regression. Do not transform the variables. Provide and interpret the F statistics, \(R^2\), standard error,and p-values only. Discuss whether the assumptions of simple linear regression met.

Scatterplot of LifeExp~TotExp

Perform Linear Regression

## 
## Call:
## lm(formula = LifeExp ~ TotExp, data = who_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -24.764  -4.778   3.154   7.116  13.292 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 6.475e+01  7.535e-01  85.933  < 2e-16 ***
## TotExp      6.297e-05  7.795e-06   8.079 7.71e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.371 on 188 degrees of freedom
## Multiple R-squared:  0.2577, Adjusted R-squared:  0.2537 
## F-statistic: 65.26 on 1 and 188 DF,  p-value: 7.714e-14

F-statistic is 65.26 and p-value is close to 0 (7.714e-14). It is likely that the model is explaining the data well, Due to the \(R^2\) value 0.2577, we can conclude that only 25% of the variation can be explained by our data. Standard error is very low. The assumptions of of simple linear regression are met.

2. Raise life expectancy to the 4.6 power (i.e., \(LifeExp^{4.6}\)). Raise total expenditures to the 0.06 power (nearly a log transform, \(TotExp^{.06}\)). Plot \(LifeExp^{4.6}\) as a function of \(TotExp^{.06}\), and r re-run the simple regression model using the transformed variables. Provide and interpret the F statistics, \(R^2\), standard error, and p-values. Which model is “better?”

Adding Column LifeExp_4.6

Adding Column TotExp_0.06

##               Country LifeExp InfantSurvival Under5Survival  TBFree      PropMD
## 1         Afghanistan      42          0.835          0.743 0.99769 0.000228841
## 2             Albania      71          0.985          0.983 0.99974 0.001143127
## 3             Algeria      71          0.967          0.962 0.99944 0.001060478
## 4             Andorra      82          0.997          0.996 0.99983 0.003297297
## 5              Angola      41          0.846          0.740 0.99656 0.000070400
## 6 Antigua and Barbuda      73          0.990          0.989 0.99991 0.000142857
##        PropRN PersExp GovtExp TotExp LifeExp_4.6 TotExp_0.06
## 1 0.000572294      20      92    112    29305338    1.327251
## 2 0.004614439     169    3128   3297   327935478    1.625875
## 3 0.002091362     108    5184   5292   327935478    1.672697
## 4 0.003500000    2589  169725 172314   636126841    2.061481
## 5 0.001146162      36    1620   1656    26230450    1.560068
## 6 0.002773810     503   12543  13046   372636298    1.765748

Scatterplot of LifeExp_4.6~TotExp_0.06

Perform Linear Regression

## 
## Call:
## lm(formula = LifeExp_4.6 ~ TotExp_0.06, data = who_data)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -308616089  -53978977   13697187   59139231  211951764 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -736527910   46817945  -15.73   <2e-16 ***
## TotExp_0.06  620060216   27518940   22.53   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 90490000 on 188 degrees of freedom
## Multiple R-squared:  0.7298, Adjusted R-squared:  0.7283 
## F-statistic: 507.7 on 1 and 188 DF,  p-value: < 2.2e-16

From this above, the linear relationship looks very clear between these 2 variables.

Also as we see the R squared values are high to assume that this is a good fit for the data.

We see that the new model fits better as compared to the model we built earlier.

3.Using the results from 2, forecast life expectancy when TotExp^.06 =1.5. Then forecast life expectancy when TotExp^.06=2.5.

Forecast life expectancy when \(TotExp^{.06}\)=1.5. Then forecast life expectancy when \(TotExp^{.06}\)=2.5

##        1 
## 63.31153
##        1 
## 86.50645

4. Build the following multiple regression model and interpret the F Statistics, R^2, standard error, and p-values. How good is the model?

LifeExp = b0+b1 x PropMd + b2 x TotExp +b3 x PropMD x TotExp

## 
## Call:
## lm(formula = LifeExp ~ PropMD + TotExp + PropMD * TotExp, data = who_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -27.320  -4.132   2.098   6.540  13.074 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    6.277e+01  7.956e-01  78.899  < 2e-16 ***
## PropMD         1.497e+03  2.788e+02   5.371 2.32e-07 ***
## TotExp         7.233e-05  8.982e-06   8.053 9.39e-14 ***
## PropMD:TotExp -6.026e-03  1.472e-03  -4.093 6.35e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.765 on 186 degrees of freedom
## Multiple R-squared:  0.3574, Adjusted R-squared:  0.3471 
## F-statistic: 34.49 on 3 and 186 DF,  p-value: < 2.2e-16

\(R^2\) value is low so the model in not a great fit.

5. Forecast LifeExp when PropMD=.03 and TotExp = 14. Does this forecast seem realistic? Why or why not?

##       1 
## 107.696

The forecast does not seem to be realistic as it is a very high value of life expectancy which is not very very rare.