Inference for numerical data

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 notebook.

# for Mac OS users only!
# if you run into any SSL certification issues, 
# you may need to run the following command for a Mac OS installation.
# $/Applications/Python 3.x/Install Certificates.command
# if this does not fix the issue, run this command instead.
import os, ssl
if (not os.environ.get('PYTHONHTTPSVERIFY', '') and
    getattr(ssl, '_create_unverified_context', None)): 
    ssl._create_default_https_context = ssl._create_unverified_context

import pandas as pd

nc = pd.read_csv('https://www.openintro.org/stat/data/nc.csv')

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?

As a first step in the analysis, we should consider summaries of the data. This can be done using the describe() and info():

nc.describe()

	fage	mage	weeks	visits	gained	weight
count	829.000000	1000.000000	998.000000	991.000000	973.000000	1000.00000
mean	30.255730	27.000000	38.334669	12.104945	30.325797	7.10100
std	6.763766	6.213583	2.931553	3.954934	14.241297	1.50886
min	14.000000	13.000000	20.000000	0.000000	0.000000	1.00000
25%	25.000000	22.000000	37.000000	10.000000	20.000000	6.38000
50%	30.000000	27.000000	39.000000	12.000000	30.000000	7.31000
75%	35.000000	32.000000	40.000000	15.000000	38.000000	8.06000
max	55.000000	50.000000	45.000000	30.000000	85.000000	11.75000

nc.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 1000 entries, 0 to 999
Data columns (total 13 columns):
fage              829 non-null float64
mage              1000 non-null int64
mature            1000 non-null object
weeks             998 non-null float64
premie            998 non-null object
visits            991 non-null float64
marital           999 non-null object
gained            973 non-null float64
weight            1000 non-null float64
lowbirthweight    1000 non-null object
gender            1000 non-null object
habit             999 non-null object
whitemom          998 non-null object
dtypes: float64(5), int64(1), object(7)
memory usage: 101.6+ KB

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.

Exercise 2

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

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 mean().

nc.groupby(['habit'])['weight'].mean()

habit
nonsmoker    7.144273
smoker       6.828730
Name: weight, dtype: float64

There is an observed difference, but is this difference statistically significant? In order to answer this question we will conduct a hypothesis test .

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 groupby command above but replacing mean with size.

Exercise 4

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

We will now conduct hypothesis tests for testing if the average weights of babies born to smoking and non-smoking mothers are different. For this task, we can use statsmodels, a Python module that provides classes and functions for the estimation of many different statistical models, as well as for conducting statistical tests, and statistical data exploration.

import statsmodels.stats.weightstats as st

nc_weightANDsmoker = nc[nc['habit'] == 'smoker']['weight']
nc_weightANDnonsmoker = nc[nc['habit'] == 'nonsmoker']['weight']

dsw1 = st.DescrStatsW(nc_weightANDsmoker)
dsw2 = st.DescrStatsW(nc_weightANDnonsmoker)
cm = st.CompareMeans(dsw1, dsw2)

# calculate number of observations, mean and standard deviation for each group
n_smoker = dsw1.nobs
n_nonsmoker = dsw2.nobs
mean_smoker = dsw1.mean
mean_nonsmoker = dsw2.mean
sd_smoker = dsw1.std
sd_nonsmoker = dsw2.std
print(f'n_smoker = {n_smoker}')
print(f'mean_smoker = {mean_smoker}')
print(f'sd_smoker = {sd_smoker}')
print()
print(f'n_nonsmoker = {n_nonsmoker}')
print(f'mean_nonsmoker = {mean_nonsmoker}')
print(f'sd_nonsmoker = {sd_nonsmoker}')
print()

# conduct hypothesis test
ht = cm.ztest_ind(alternative = 'two-sided', usevar = 'unequal', value = 0)

# calculate and print the standard error, the Z-score, and p-value for the hypothesis test
se = cm.std_meandiff_separatevar
testZ = ht[0]
p_value = ht[1]
print(f'Standard error = {se}')
print(f'Test statistic: Z = {testZ}')
print(f'p-value = {p_value}')

# reject or accept null hypothesis
if(p_value) < 0.05:
    print('reject null hypothesis')
else:
    print('accept null hypothesis')

n_smoker = 126.0
mean_smoker = 6.828730158730159
sd_smoker = 1.3806681061171728

n_nonsmoker = 873.0
mean_nonsmoker = 7.1442726231386
sd_nonsmoker = 1.5178105512705897

Standard error = 0.13376049190705977
Test statistic: Z = -2.359010944933488
p-value = 0.018323715325166846
reject null hypothesis

Exercise 5

Construct a confidence interval for the difference between the weights of babies born to smoking and non-smoking mothers.

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On Your Own

1. 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.
2. Calculate a new confidence interval for the same parameter at the 90% confidence level.
3. Conduct a hypothesis test evaluating whether the average weight gained by younger mothers is different than the average weight gained by mature mothers.
4. 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.
5. 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 with Python, report the statistical results, and also provide an explanation in plain language.

This lab was adapted by Vural Aksakalli and Imran Ture from OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel.

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