Data606 - Lab 7 - Inference for numerical data
Vinayak Kamath
3/21/2020
North Carolina births
In 2004, the state of North Carolina released a large data set containing information on births recorded in this state. This data set is useful to researchers studying the relation between habits and practices of expectant mothers and the birth of their children. We will work with a random sample of observations from this data set.
Exploratory analysis
Load the nc data set into our workspace.
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. |
What are the cases in this data set? How many cases are there in our sample?
Answer: The cases represents birth of a child/baby.
## [1] "There are 1000 cases in our sample"
As a first step in the analysis, we should consider summaries of the data. This can be done using the summary command:
## 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
##
##
##
## As you review the variable summaries, consider which variables are categorical and which are numerical. For numerical variables, are there outliers? If you aren’t sure or want to take a closer look at the data, make a graph.
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.
Make a side-by-side boxplot of
habitandweight. What does the plot highlight about the relationship between these two variables?Answer: The Plot below highligts the weight range for babies from non-smoker mother is larger; Also the median weight of babies from non-smoker mother is slightly higher.
The box plots show how the medians of the two distributions compare, but we can also compare the means of the distributions using the following function to split the weight variable into the habit groups, then take the mean of each using the mean function.
## nc$habit: nonsmoker
## [1] 7.144273
## ------------------------------------------------------------
## nc$habit: smoker
## [1] 6.82873There is an observed difference, but is this difference statistically significant? In order to answer this question we will conduct a hypothesis test .
Inference
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
bycommand above but replacingmeanwithlength.## nc$habit: nonsmoker ## [1] 873 ## ------------------------------------------------------------ ## nc$habit: smoker ## [1] 126Write the hypotheses for testing if the average weights of babies born to smoking and non-smoking mothers are different.
Answer: The hypotheses for the testing are:
H0: Average/Mean Weight of babies born to Non-Smoking mothers is equal to Average/Mean Weight of babies born to Smoking mothers.
HA: Average/Mean Weight of babies born to Non-Smoking mothers is NOT equal to Average/Mean Weight of babies born to Smoking mothers.
Next, we introduce a new function, inference, that we will use for conducting hypothesis tests and constructing confidence intervals.
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
Let’s pause for a moment to go through the arguments of this custom function. The first argument is y, which is the response variable that we are interested in: nc$weight. The second argument is the explanatory variable, x, which is the variable that splits the data into two groups, smokers and non-smokers: nc$habit. The third argument, est, is the parameter we’re interested in: "mean" (other options are "median", or "proportion".) Next we decide on the type of inference we want: a hypothesis test ("ht") or a confidence interval ("ci"). When performing a hypothesis test, we also need to supply the null value, which in this case is 0, since the null hypothesis sets the two population means equal to each other. The alternative hypothesis can be "less", "greater", or "twosided". Lastly, the method of inference can be "theoretical" or "simulation" based.
Change the
typeargument 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")## 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 )
By default the function reports an interval for (\(\mu_{nonsmoker} - \mu_{smoker}\)) . We can easily change this order by using the order argument:
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
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 thexvariable from the function.## Single mean ## Summary statistics:
## mean = 38.3347 ; sd = 2.9316 ; n = 998 ## Standard error = 0.0928 ## 95 % Confidence interval = ( 38.1528 , 38.5165 )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.## Single mean ## Summary statistics:## mean = 38.3347 ; sd = 2.9316 ; n = 998 ## Standard error = 0.0928 ## 90 % Confidence interval = ( 38.182 , 38.4873 )Conduct a hypothesis test evaluating whether the average weight gained by younger mothers is different than the average weight gained by mature mothers.
Answer: The hypotheses for the testing are:
H0: Average/Mean Weight gained by younger mothers is equal to Average/Mean Weight gained by mature mothers.
HA: Average/Mean Weight gained by younger mothers is NOT equal to Average/Mean Weight gained by mature mothers.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
==> NULL hypotheses cannot be rejected as p-value > 0.05.
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.
# Using the by fucntion to get the min and max of mother age by maturity by(nc$mage, nc$mature, min )## nc$mature: mature mom ## [1] 35 ## ------------------------------------------------------------ ## nc$mature: younger mom ## [1] 13## nc$mature: mature mom ## [1] 50 ## ------------------------------------------------------------ ## nc$mature: younger mom ## [1] 34==> The Age Cut off would be at 34 and lower for younger mom; 35 and higher for mature mom
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
inferencefunction, report the statistical results, and also provide an explanation in plain language.Answer: The hypotheses for the testing are:
H0: Average/Mean visits by whitemom mothers is equal to Average/Mean visits by not white mothers.
HA: Average/Mean visits by whitemom mothers is NOT equal to Average/Mean visits by not white mothers.inference(y = nc$visits, x = nc$whitemom, est = "mean", type = "ht", null = 0, alternative = "twosided", method = "theoretical")## Response variable: numerical, Explanatory variable: categorical ## Difference between two means ## Summary statistics: ## n_not white = 279, mean_not white = 11.6272, sd_not white = 4.3644 ## n_white = 710, mean_white = 12.3014, sd_white = 3.7701## Observed difference between means (not white-white) = -0.6742 ## ## H0: mu_not white - mu_white = 0 ## HA: mu_not white - mu_white != 0 ## Standard error = 0.297 ## Test statistic: Z = -2.269 ## p-value = 0.0232
==> NULL hypotheses can be rejected as p-value < 0.05.