Inference for numerical data

Exploratory analysis

Load the nc data set into our workspace.

download.file("http://www.openintro.org/stat/data/nc.RData", destfile = "nc.RData")
load("nc.RData")

We have observations on 13 different variables, some categorical and some numerical. The meaning of each variable is as follows.

variable	description
`fage`	father’s age in years.
`mage`	mother’s age in years.
`mature`	maturity status of mother.
`weeks`	length of pregnancy in weeks.
`premie`	whether the birth was classified as premature (premie) or full-term.
`visits`	number of hospital visits during pregnancy.
`marital`	whether mother is `married` or `not married` at birth.
`gained`	weight gained by mother during pregnancy in pounds.
`weight`	weight of the baby at birth in pounds.
`lowbirthweight`	whether baby was classified as low birthweight (`low`) or not (`not low`).
`gender`	gender of the baby, `female` or `male`.
`habit`	status of the mother as a `nonsmoker` or a `smoker`.
`whitemom`	whether mom is `white` or `not white`.

Exercise 1. What are the cases in this data set? How many cases are there in our sample?

The cases are mothers who have had babies. There are 1000 cases.

summary(nc)

##       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  
##                 
##                 
##                 
##

As you review the variable summaries, consider which variables are categorical and which are numerical.

Categorical: Maturity, premie, marital, lowbirthweight, gender, habit. whitemom

Numerical: fage, mage, weeks,gained, visits, weight.

For numerical variables, are there outliers?

par(mfrow=c(3,2))
hist(nc$fage)
hist(nc$mage)
hist(nc$weeks)
hist(nc$gained)
hist(nc$visits)
hist(nc$weight)

You can see that weight and weeks have low outliers and visits has high ourliers.

Consider the possible relationship between a mother’s smoking habit and the weight of her baby. Plotting the data is a useful first step because it helps us quickly visualize trends, identify strong associations, and develop research questions.

Exercise 2

boxplot(nc$weight ~ nc$habit, data = nc, col = "lightblue")

Smokers appear to have babies with less weight.

by(nc$weight, nc$habit, mean)

## nc$habit: nonsmoker ## [1] 7.144273 ## -------------------------------------------------------- ## nc$habit: smoker ## [1] 6.82873

Excercise 3

len<-by(nc$weight, nc$habit, length) len

## nc$habit: nonsmoker ## [1] 873 ## -------------------------------------------------------- ## nc$habit: smoker ## [1] 126

len>30

## nc$habit ## nonsmoker smoker ## TRUE TRUE

Both are true so we have conditions for inference.

Exercise 4

H0: There is no difference in baby’s weights between the means for smokers and nonsmokers H1: There is a difference in mean baby weights between the groups smoking and non smoking (tail=2)

library(openintro)

## Warning: package 'openintro' was built under R version 3.2.5

## Please visit openintro.org for free statistics materials

## ## Attaching package: 'openintro'

## The following object is masked _by_ '.GlobalEnv': ## ## normTail

## The following object is masked from 'package:datasets': ## ## cars

library(BHH2)

## Warning: package 'BHH2' was built under R version 3.2.5

## ## Attaching package: 'BHH2'

## The following object is masked from 'package:openintro': ## ## dotPlot

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

Exercise 5

inference(y = nc$weight, x = nc$habit, est = "mean", type = "ci", null = 0, alternative = "twosided", method = "theoretical", order=c("smoker","nonsmoker"))

## Response variable: numerical, Explanatory variable: categorical ## Difference between two means ## Summary statistics: ## n_smoker = 126, mean_smoker = 6.8287, sd_smoker = 1.3862 ## n_nonsmoker = 873, mean_nonsmoker = 7.1443, sd_nonsmoker = 1.5187

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

Problem 1

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 )

Problem 2

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 )

Problem 3

inference(y = nc$gained,x=nc$mature, est = "mean", type = "ht", null = 0, alternative = "twosided", method = "theoretical",conflevel = 0.95)

## 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

Not significantly associated

Problem 4

mat<-subset(nc,nc$mature=="mature mom") un<-subset(nc,nc$mature!="mature mom") ##par(mfrow=c(1,2)) u<-hist(mat$mage,col="red")

m<-hist(un$mage,col="blue")

plot(u,xlim = c(14,60),col="red") lines(m,col="green")

max(un$mage)

## [1] 34

min(mat$mage)

## [1] 35

So the natural cutoff is below 35 for not mature and 35 and above for mature.

Problem 5

H0: There is no difference in term mean length in weeks between white and non white mothers. H1: There is a difference in mean term length between white and non white mothers.

inference(y = nc$weeks,x=nc$whitemom, est = "mean", type = "ht", null = 0, alternative = "twosided", method = "theoretical",conflevel = 0.95)

## Response variable: numerical, Explanatory variable: categorical ## Difference between two means ## Summary statistics: ## n_not white = 284, mean_not white = 37.8768, sd_not white = 3.6705 ## n_white = 712, mean_white = 38.5126, sd_white = 2.5617

## Observed difference between means (not white-white) = -0.6359 ## ## H0: mu_not white - mu_white = 0 ## HA: mu_not white - mu_white != 0 ## Standard error = 0.238 ## Test statistic: Z = -2.672 ## p-value = 0.0076

p=0.0076

So the mean of not white is 0.6358 weeks less and this difference is signficant.

Inference for numerical data

Scott Ogden

North Carolina births

Exploratory analysis

Exercise 2

Excercise 3

Exercise 4

Exercise 5

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5