Wk. 8 RLab

Inference for Numerical Data

North Carolina Births

Exploratory Analysis

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

Exercise 1

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

summary(nc)

##       fage            mage            mature        weeks             premie   
##  Min.   :14.00   Min.   :13   mature mom :133   Min.   :20.00   full term:846  
##  1st Qu.:25.00   1st Qu.:22   younger mom:867   1st Qu.:37.00   premie   :152  
##  Median :30.00   Median :27                     Median :39.00   NA's     :  2  
##  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                      
##      visits            marital        gained          weight      
##  Min.   : 0.0   married    :386   Min.   : 0.00   Min.   : 1.000  
##  1st Qu.:10.0   not married:613   1st Qu.:20.00   1st Qu.: 6.380  
##  Median :12.0   NA's       :  1   Median :30.00   Median : 7.310  
##  Mean   :12.1                     Mean   :30.33   Mean   : 7.101  
##  3rd Qu.:15.0                     3rd Qu.:38.00   3rd Qu.: 8.060  
##  Max.   :30.0                     Max.   :85.00   Max.   :11.750  
##  NA's   :9                        NA's   :27                      
##  lowbirthweight    gender          habit          whitemom  
##  low    :111    female:503   nonsmoker:873   not white:284  
##  not low:889    male  :497   smoker   :126   white    :714  
##                              NA's     :  1   NA's     :  2  
##                                                             
##                                                             
##                                                             
##

head(nc)

##   fage mage      mature weeks    premie visits marital gained weight
## 1   NA   13 younger mom    39 full term     10 married     38   7.63
## 2   NA   14 younger mom    42 full term     15 married     20   7.88
## 3   19   15 younger mom    37 full term     11 married     38   6.63
## 4   21   15 younger mom    41 full term      6 married     34   8.00
## 5   NA   15 younger mom    39 full term      9 married     27   6.38
## 6   NA   15 younger mom    38 full term     19 married     22   5.38
##   lowbirthweight gender     habit  whitemom
## 1        not low   male nonsmoker not white
## 2        not low   male nonsmoker not white
## 3        not low female nonsmoker     white
## 4        not low   male nonsmoker     white
## 5        not low female nonsmoker not white
## 6            low   male nonsmoker not white

nrow(nc)

## [1] 1000

Answer: The cases study details of births in North Carolina. There were 1000 sample cases in this data set.

hist(nc$fage)

hist(nc$mage)

hist(nc$weeks)

hist(nc$gained)

hist(nc$weight)

The following variables are numeric:

Age (mother and father), length of pregnancy, visits, gained, and weight.

The following variables are categorical:

Mature, premie, marital, lowbirthweight, gender, habit, and white mom.

Exercise 2

Make a side-by-side boxplot of habit and weight. What does the plot highlight about the relationship between these two variables?

# Boxplot of habit and weight 
boxplot(weight~habit,data=nc, main="Relation Between Mother's Habit and Baby's Weight", 
    ylab="Baby's Weight", xlab="Mother Smoker/Non-Smoker")

summary(nc$habit)

## nonsmoker    smoker      NA's 
##       873       126         1

summary(nc$weight, nc$habit$smoker)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.000   6.380   7.310   7.101   8.060  11.750

summary(nc$weight, nc$habit$nonsmoker)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.000   6.380   7.310   7.101   8.060  11.750

Answer: The plot illustrates how the baby’s weight compares between babies with a mother who is a smoker versus nonsmoker. The median birth weight for babies with a non-smoking mother is greater than the birthweights for babies with a smoking mother.

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

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

Inference

Exercise 3

Check if the conditions necessary for inference are satisfied. Note that you will need to obtain sample sizes to check the conditions. You can compute the group size using the same by command above but replacing mean with length.

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

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

Exercise 4

Write the hypotheses for testing if the average weights of babies born to smoking and non-smoking mothers are different.

Answer: The hypothesis for testing if the average weights of babies born to smoking and non-smoking mothers are different would be:

Null Hypothesis: The average weights of babies born to smoking mothers is equal to non-smoking mothers.

Alternative Hypothesis: The average weights of babes born to smoking mothers is not equal to non-smoking mothers.

inference(y = nc$weight, x = nc$habit, est = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical")

## Warning: package 'BHH2' was built under R version 3.6.3

## 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

Change the type argument to “ci” to construct and record a confidence interval for the difference between the weights of babies born to smoking and non-smoking mothers.

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 )

On Your Own

Exercise 6

Calculate a 95% confidence interval for the average length of pregnancies (weeks) and interpret it in context. Note that since you’re doing inference on a single population parameter, there is no explanatory variable, so you can omit the x variable from the function.

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 )

Exercise 7

Calculate a new confidence interval for the same parameter at the 90% confidence level. You can change the confidence level by adding a new argument to the function: conflevel = 0.90.

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 )

Exercise 8

*Conduct a hypothesis test evaluating whether the average weight gained by younger mothers is different than the average weight gained by mature mothers.

Answer: Null Hypothesis: There is no difference in the mean of the birth weight between younger and mature mothers who don’t smoke

Alternative Hypothesis: There is a difference in the mean of the birth weight between younger and mature mothers

inference(y = nc$weight, x = nc$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 = 133, mean_mature mom = 7.1256, sd_mature mom = 1.6591
## n_younger mom = 867, mean_younger mom = 7.0972, sd_younger mom = 1.4855

## Observed difference between means (mature mom-younger mom) = 0.0283
## 
## H0: mu_mature mom - mu_younger mom = 0 
## HA: mu_mature mom - mu_younger mom != 0 
## Standard error = 0.152 
## Test statistic: Z =  0.186 
## p-value =  0.8526

Answer: Based on statistical data, including a P-value of 0.85, there is very weak evidence to reject the null hypothesis.

Exercise 9

Now, a non-inference task: Determine the age cutoff for younger and mature mothers. Use a method of your choice, and explain how your method works.

install.packages("dplyr")

## Installing package into 'C:/Users/Valued Customer/Documents/R/win-library/3.6'
## (as 'lib' is unspecified)

## package 'dplyr' successfully unpacked and MD5 sums checked

## Warning: cannot remove prior installation of package 'dplyr'

## Warning in file.copy(savedcopy, lib, recursive = TRUE):
## problem copying C:\Users\Valued Customer\Documents\R\win-
## library\3.6\00LOCK\dplyr\libs\x64\dplyr.dll to C:\Users\Valued
## Customer\Documents\R\win-library\3.6\dplyr\libs\x64\dplyr.dll: Permission denied

## Warning: restored 'dplyr'

## 
## The downloaded binary packages are in
##  C:\Users\Valued Customer\AppData\Local\Temp\RtmpIlusdf\downloaded_packages

library(dplyr)

## Warning: package 'dplyr' was built under R version 3.6.3

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(dplyr)
nc %>% group_by(mature) %>% summarize(max(mage))

## # A tibble: 2 x 2
##   mature      `max(mage)`
##   <fct>             <int>
## 1 mature mom           50
## 2 younger mom          34

nc %>% group_by(mature) %>% summarize(min(mage))

## # A tibble: 2 x 2
##   mature      `min(mage)`
##   <fct>             <int>
## 1 mature mom           35
## 2 younger mom          13

Answer: The data states that young moms are age 34 or younger. Mature moms are age 35 and older. The oldest mother included ws 50 years old. The youngest mom included was 13 years old.

Exercise 10

Pick a pair of numerical and categorical variables and come up with a research question evaluating the relationship between these variables. Formulate the question in a way that it can be answered using a hypothesis test and/or a confidence interval. Answer your question using the inference function, report the statistical results, and also provide an explanation in plain language.

Answer:

Research question: Does the mother’s smoking habit affect weight gained by mothers during pregnancy?

H0: µ{smoker} - µ{nonsmoker} = 0, There is no difference in the mean of the weight gained during pregnancy of between married and unmarried mothers

HA: µ{smoker} - µ{not nonsmoker} != 0, There is a difference in the mean of the weight gained during pregnancy between smoker and nonsmoker mothers

inference(y = nc$gained, x = nc$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 = 851, mean_nonsmoker = 30.0964, sd_nonsmoker = 14.0226
## n_smoker = 122, mean_smoker = 31.9262, sd_smoker = 15.6512

## Observed difference between means (nonsmoker-smoker) = -1.8299
## 
## H0: mu_nonsmoker - mu_smoker = 0 
## HA: mu_nonsmoker - mu_smoker != 0 
## Standard error = 1.496 
## Test statistic: Z =  -1.223 
## p-value =  0.2214

inference(y = nc$gained, x = nc$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 = 851, mean_nonsmoker = 30.0964, sd_nonsmoker = 14.0226
## n_smoker = 122, mean_smoker = 31.9262, sd_smoker = 15.6512

## Observed difference between means (nonsmoker-smoker) = -1.8299
## 
## Standard error = 1.4963 
## 95 % Confidence interval = ( -4.7626 , 1.1028 )

Answer: Based on the hypothesis test and confidence interval inference tests, we cannot reject the null hypothesis. We then conclude that there is no evidence based on statistical data to show that there is a difference between the weight gained by smoker and nonsmoker mothers during pregnancy.

Wk. 8 RLab

Jennifer Johnson

29/03/2020

Inference for Numerical Data

North Carolina Births

Exploratory Analysis

Exercise 1

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

Answer: The cases study details of births in North Carolina. There were 1000 sample cases in this data set.

The following variables are numeric:

Age (mother and father), length of pregnancy, visits, gained, and weight.

The following variables are categorical:

Mature, premie, marital, lowbirthweight, gender, habit, and white mom.

Exercise 2

Make a side-by-side boxplot of habit and weight. What does the plot highlight about the relationship between these two variables?

Answer: The plot illustrates how the baby’s weight compares between babies with a mother who is a smoker versus nonsmoker. The median birth weight for babies with a non-smoking mother is greater than the birthweights for babies with a smoking mother.

Inference

Exercise 3

Check if the conditions necessary for inference are satisfied. Note that you will need to obtain sample sizes to check the conditions. You can compute the group size using the same by command above but replacing mean with length.

Exercise 4

Write the hypotheses for testing if the average weights of babies born to smoking and non-smoking mothers are different.

Answer: The hypothesis for testing if the average weights of babies born to smoking and non-smoking mothers are different would be:

Null Hypothesis: The average weights of babies born to smoking mothers is equal to non-smoking mothers.

Alternative Hypothesis: The average weights of babes born to smoking mothers is not equal to non-smoking mothers.

Exercise 5

Change the type argument to “ci” to construct and record a confidence interval for the difference between the weights of babies born to smoking and non-smoking mothers.

On Your Own

Exercise 6

Calculate a 95% confidence interval for the average length of pregnancies (weeks) and interpret it in context. Note that since you’re doing inference on a single population parameter, there is no explanatory variable, so you can omit the x variable from the function.

Exercise 7

Calculate a new confidence interval for the same parameter at the 90% confidence level. You can change the confidence level by adding a new argument to the function: conflevel = 0.90.

Exercise 8

*Conduct a hypothesis test evaluating whether the average weight gained by younger mothers is different than the average weight gained by mature mothers.

Answer: Null Hypothesis: There is no difference in the mean of the birth weight between younger and mature mothers who don’t smoke

Alternative Hypothesis: There is a difference in the mean of the birth weight between younger and mature mothers

Answer: Based on statistical data, including a P-value of 0.85, there is very weak evidence to reject the null hypothesis.

Exercise 9

Now, a non-inference task: Determine the age cutoff for younger and mature mothers. Use a method of your choice, and explain how your method works.

Answer: The data states that young moms are age 34 or younger. Mature moms are age 35 and older. The oldest mother included ws 50 years old. The youngest mom included was 13 years old.

Exercise 10

Answer:

Research question: Does the mother’s smoking habit affect weight gained by mothers during pregnancy?

H0: µ{smoker} - µ{nonsmoker} = 0, There is no difference in the mean of the weight gained during pregnancy of between married and unmarried mothers

HA: µ{smoker} - µ{not nonsmoker} != 0, There is a difference in the mean of the weight gained during pregnancy between smoker and nonsmoker mothers

Answer: Based on the hypothesis test and confidence interval inference tests, we cannot reject the null hypothesis. We then conclude that there is no evidence based on statistical data to show that there is a difference between the weight gained by smoker and nonsmoker mothers during pregnancy.