Inference for numerical data

Our study looks at 1000 randomly sampled cases of births in North Carolina in 2004. 

## 'data.frame':    1000 obs. of  13 variables:
##  $ fage          : int  NA NA 19 21 NA NA 18 17 NA 20 ...
##  $ mage          : int  13 14 15 15 15 15 15 15 16 16 ...
##  $ mature        : Factor w/ 2 levels "mature mom","younger mom": 2 2 2 2 2 2 2 2 2 2 ...
##  $ weeks         : int  39 42 37 41 39 38 37 35 38 37 ...
##  $ premie        : Factor w/ 2 levels "full term","premie": 1 1 1 1 1 1 1 2 1 1 ...
##  $ visits        : int  10 15 11 6 9 19 12 5 9 13 ...
##  $ marital       : Factor w/ 2 levels "married","not married": 1 1 1 1 1 1 1 1 1 1 ...
##  $ gained        : int  38 20 38 34 27 22 76 15 NA 52 ...
##  $ weight        : num  7.63 7.88 6.63 8 6.38 5.38 8.44 4.69 8.81 6.94 ...
##  $ lowbirthweight: Factor w/ 2 levels "low","not low": 2 2 2 2 2 1 2 1 2 2 ...
##  $ gender        : Factor w/ 2 levels "female","male": 2 2 1 2 1 2 2 2 2 1 ...
##  $ habit         : Factor w/ 2 levels "nonsmoker","smoker": 1 1 1 1 1 1 1 1 1 1 ...
##  $ whitemom      : Factor w/ 2 levels "not white","white": 1 1 2 2 1 1 1 1 2 2 ...
##       fage            mage            mature        weeks      
##  Min.   :14.00   Min.   :13   mature mom :133   Min.   :20.00  
##  1st Qu.:25.00   1st Qu.:22   younger mom:867   1st Qu.:37.00  
##  Median :30.00   Median :27                     Median :39.00  
##  Mean   :30.26   Mean   :27                     Mean   :38.33  
##  3rd Qu.:35.00   3rd Qu.:32                     3rd Qu.:40.00  
##  Max.   :55.00   Max.   :50                     Max.   :45.00  
##  NA's   :171                                    NA's   :2      
##        premie        visits            marital        gained     
##  full term:846   Min.   : 0.0   married    :386   Min.   : 0.00  
##  premie   :152   1st Qu.:10.0   not married:613   1st Qu.:20.00  
##  NA's     :  2   Median :12.0   NA's       :  1   Median :30.00  
##                  Mean   :12.1                     Mean   :30.33  
##                  3rd Qu.:15.0                     3rd Qu.:38.00  
##                  Max.   :30.0                     Max.   :85.00  
##                  NA's   :9                        NA's   :27     
##      weight       lowbirthweight    gender          habit    
##  Min.   : 1.000   low    :111    female:503   nonsmoker:873  
##  1st Qu.: 6.380   not low:889    male  :497   smoker   :126  
##  Median : 7.310                               NA's     :  1  
##  Mean   : 7.101                                              
##  3rd Qu.: 8.060                                              
##  Max.   :11.750                                              
##                                                              
##       whitemom  
##  not white:284  
##  white    :714  
##  NA's     :  2  
##                 
##                 
##                 
##

Smokers in North Carolina in our subset of 2004 births had babies that were slightly smaller, on average. There is a curious pattern of low birth-weight outliers in the nonsmoker category that doesn’t show up for smokers. The reason for this, and whether it exists in any broader population, could be a fruitful topic for future research. It might lend itsself well to machine learning classifiers. 

by(nc$weight, nc$habit, mean)
## nc$habit: nonsmoker
## [1] 7.144273
## -------------------------------------------------------- 
## nc$habit: smoker
## [1] 6.82873
by(nc$weight, nc$habit, length)
## nc$habit: nonsmoker
## [1] 873
## -------------------------------------------------------- 
## nc$habit: smoker
## [1] 126
library(e1071)                    
test.for.nonsmoker.skew.n<-subset(nc, nc$habit=="nonsmoker")
skewness(test.for.nonsmoker.skew.n$weight)
## [1] -1.182798
test.for.nonsmoker.skew.s<-subset(nc, nc$habit=="smoker")
skewness(test.for.nonsmoker.skew.s$weight)
## [1] -0.9898247

The conditions for inference are not met. The observations are randomly chosen and independent. The sample size is large enough for inference, but not over 10%. They both, however, are quite skewed.

H0 : The birth weights for babies born to smokers are the same.
H1 : The birth weights for babies born to smokers are different than for those born to non-smokers. 

inference(y = nc$weight, x = nc$habit, est = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical")
## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_nonsmoker = 873, mean_nonsmoker = 7.1443, sd_nonsmoker = 1.5187
## n_smoker = 126, mean_smoker = 6.8287, sd_smoker = 1.3862
## Observed difference between means (nonsmoker-smoker) = 0.3155
## 
## H0: mu_nonsmoker - mu_smoker = 0 
## HA: mu_nonsmoker - mu_smoker != 0 
## Standard error = 0.134 
## Test statistic: Z =  2.359 
## p-value =  0.0184

inference(y = nc$weight, x = nc$habit, est = "mean", type = "ci", null = 0, 
          alternative = "twosided", method = "theoretical")
## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_nonsmoker = 873, mean_nonsmoker = 7.1443, sd_nonsmoker = 1.5187
## n_smoker = 126, mean_smoker = 6.8287, sd_smoker = 1.3862

## Observed difference between means (nonsmoker-smoker) = 0.3155
## 
## Standard error = 0.1338 
## 95 % Confidence interval = ( 0.0534 , 0.5777 )

On your own

The 95% confidence interval for weeks’ gestation is from 38.1528 to 38.5165. That means we expect the mean of the population to be in this interval, with 95% confidence. 

inference(y = nc$weeks, est = "mean", type = "ci", null = 0, 
          alternative = "twosided", method = "theoretical")
## Single mean 
## Summary statistics:

## mean = 38.3347 ;  sd = 2.9316 ;  n = 998 
## Standard error = 0.0928 
## 95 % Confidence interval = ( 38.1528 , 38.5165 )

The 90% confidence interval for weeks’ gestation is from 38.182 to 38.4873. That means we expect the mean of the population to be in this interval, with 90% confidence. It is only slightly narrower than our 95% c.i. 

inference(y = nc$weeks, est = "mean", type = "ci", null = 0, 
          alternative = "twosided", method = "theoretical",conflevel = 0.90)
## Single mean 
## Summary statistics:

## mean = 38.3347 ;  sd = 2.9316 ;  n = 998 
## Standard error = 0.0928 
## 90 % Confidence interval = ( 38.182 , 38.4873 )

Is there a difference in weight gain between mature mothers and younger mothers?
H0 : The weight gain for older and younger mothers are the same.
H1 : The weight gain for older and younger mothers are different. 

inference(y = nc$gained, x = nc$mature, est = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical")
## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_mature mom = 129, mean_mature mom = 28.7907, sd_mature mom = 13.4824
## n_younger mom = 844, mean_younger mom = 30.5604, sd_younger mom = 14.3469
## Observed difference between means (mature mom-younger mom) = -1.7697
## 
## H0: mu_mature mom - mu_younger mom = 0 
## HA: mu_mature mom - mu_younger mom != 0 
## Standard error = 1.286 
## Test statistic: Z =  -1.376 
## p-value =  0.1686

There is insufficient evidence to show that mature mothers are likely to gain more weight during pregnancy. The p-value is .1686 

older<-subset(nc,nc$mature=="mature mom")
min(older$mage,na.rm = TRUE)
## [1] 35
younger<-subset(nc,nc$mature=="younger mom")
max(younger$mage,na.rm = TRUE)
## [1] 34

Mothers are younger if they are 34 or younger. We created a subset for both younger and mature and found a min and a max to find the boundary.

Is there a difference between babies born to mature mothers compared to younger mothers?
H0 : The birth timing for babies born to older mothers are the same.
H1 : The birth timing for babies born to older mothers are different than for those born to younger mothers. 

inference(y = nc$weeks, x = nc$mature, est = "mean", type = "ht", null = 0, 
          alternative = "less", method = "theoretical")
## Response variable: numerical, Explanatory variable: categorical
## Difference between two means
## Summary statistics:
## n_mature mom = 132, mean_mature mom = 38.0227, sd_mature mom = 3.2184
## n_younger mom = 866, mean_younger mom = 38.3822, sd_younger mom = 2.8844
## Observed difference between means (mature mom-younger mom) = -0.3595
## 
## H0: mu_mature mom - mu_younger mom = 0 
## HA: mu_mature mom - mu_younger mom < 0 
## Standard error = 0.297 
## Test statistic: Z =  -1.211 
## p-value =  0.1129

There is insufficient evidence to show that mature mothers are more likely to give birth to babies earlier in their gestation. The p-value is .1129