Complete all Exercises, and submit answers to Questions on the Coursera platform.

Getting Started

Load packages

In this lab we will explore the data using the dplyr package and visualize it using the ggplot2 package for data visualization. The data can be found in the companion package for this course, statsr.

Let’s load the packages.

library(statsr)
library(dplyr)
library(ggplot2)

The data

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.

Load the nc data set into our workspace.

data(nc)

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.
  1. There are 1,000 cases in this data set, what do the cases represent?
    1. The hospitals where the births took place
    2. The fathers of the children
    3. The days of the births
    4. The births
    The births is represented by 1000 cases in this data set.

As a first step in the analysis, we should take a look at the variables in the dataset. This can be done using the str command:

str(nc)
## tibble [1,000 x 13] (S3: tbl_df/tbl/data.frame)
##  $ fage          : int [1:1000] NA NA 19 21 NA NA 18 17 NA 20 ...
##  $ mage          : int [1:1000] 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 [1:1000] 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 [1:1000] 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 [1:1000] 38 20 38 34 27 22 76 15 NA 52 ...
##  $ weight        : num [1:1000] 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 ...

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.

Exploratory data analysis

We will first start with analyzing the weight gained by mothers throughout the pregnancy: gained.

Using visualization and summary statistics, describe the distribution of weight gained by mothers during pregnancy. The summary function can also be useful.

summary(nc$gained)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##    0.00   20.00   30.00   30.33   38.00   85.00      27
  1. How many mothers are we missing weight gain data from?
    1. 0
    2. 13
    3. 27
    4. 31
    There are 27 missing weight gained by mothers

Next, 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.

  1. Make side-by-side boxplots of habit and weight. Which of the following is false about the relationship between habit and weight?
    1. Median birth weight of babies born to non-smoker mothers is slightly higher than that of babies born to smoker mothers.
    2. Range of birth weights of babies born to non-smoker mothers is greater than that of babies born to smoker mothers.
    3. Both distributions are extremely right skewed.
    4. The IQRs of the distributions are roughly equal.
      5. Firstly, the distributions that represents in the boxplots graph, can tell the shape of the distribution namely right skew(positive skew), left skew(negative skew) and symmetric respectively. Therefore, look at the one box plot, you can see that the median is the middle line to separate the box to 2 pieces, if the box which is above the median is larger than the lower one so it’s called right skew. On the other hand, if the lower box is larger than the above one it can consequently tell that is the left skew. As the result, the distribution of the birth weights of babies born to non-smoker mothers is fairly symmetric distribution. And the distribution of the birth weights of babies born to smoker mothers is slightly left skew distribution. In brief, The False statement is Both distributions are extremely right skewed. 
# type your code for the Question 3 here, and Knit
# weight is a numeric variable VS habit as factor 
ggplot(nc, aes(x = habit, y = weight)) +
    geom_boxplot() +
    ggtitle("The relation between habit and weight")

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 to first group the data by the habit variable, and then calculate the mean weight in these groups using the mean function.

nc %>%
  group_by(habit) %>%
  summarise(mean_weight = mean(weight))
## `summarise()` ungrouping output (override with `.groups` argument)
## # A tibble: 3 x 2
##   habit     mean_weight
##   <fct>           <dbl>
## 1 nonsmoker        7.14
## 2 smoker           6.83
## 3 <NA>             3.63

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: Are all conditions necessary for inference satisfied? Comment on each. You can compute the group sizes using the same by command above but replacing mean(weight) with n().

  1. What are the hypotheses for testing if the average weights of babies born to smoking and non-smoking mothers are different?
    1. \(H_0: \mu_{smoking} = \mu_{non-smoking}\); \(H_A: \mu_{smoking} > \mu_{non-smoking}\)
    2. \(H_0: \mu_{smoking} = \mu_{non-smoking}\); \(H_A: \mu_{smoking} \ne \mu_{non-smoking}\)
    3. \(H_0: \bar{x}_{smoking} = \bar{x}_{non-smoking}\); \(H_A: \bar{x}_{smoking} > \bar{x}_{non-smoking}\)
    4. \(H_0: \bar{x}_{smoking} = \bar{x}_{non-smoking}\); \(H_A: \bar{x}_{smoking} > \bar{x}_{non-smoking}\)
    5. \(H_0: \mu_{smoking} \ne \mu_{non-smoking}\); \(H_A: \mu_{smoking} = \mu_{non-smoking}\)

    the hypotheses for testing if the average weights of babies born to smoking and non-smoking mothers are different is \(H_0: \mu_{smoking} = \mu_{non-smoking}\); \(H_A: \mu_{smoking} \ne \mu_{non-smoking}\)

Next, we introduce a new function, inference, that we will use for conducting hypothesis tests and constructing confidence intervals.

Then, run the following:

inference(y = weight, x = habit, data = nc, statistic = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical")
## Response variable: numerical
## Explanatory variable: categorical (2 levels) 
## n_nonsmoker = 873, y_bar_nonsmoker = 7.1443, s_nonsmoker = 1.5187
## n_smoker = 126, y_bar_smoker = 6.8287, s_smoker = 1.3862
## H0: mu_nonsmoker =  mu_smoker
## HA: mu_nonsmoker != mu_smoker
## t = 2.359, df = 125
## p_value = 0.0199

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: weight. The second argument is the explanatory variable, x, which is the variable that splits the data into two groups, smokers and non-smokers: habit. The third argument, data, is the data frame these variables are stored in. Next is statistic, which is the sample statistic we’re using, or similarly, the population parameter we’re estimating. In future labs we can also work with “median” and “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.

For more information on the inference function see the help file with ?inference.

Exercise: What is the conclusion of the hypothesis test?

  1. Change the type argument to "ci" to construct and record a confidence interval for the difference between the weights of babies born to nonsmoking and smoking mothers, and interpret this interval in context of the data. Note that by default you’ll get a 95% confidence interval. If you want to change the confidence level, add a new argument (conf_level) which takes on a value between 0 and 1. Also note that when doing a confidence interval arguments like null and alternative are not useful, so make sure to remove them.
    1. We are 95% confident that babies born to nonsmoker mothers are on average 0.05 to 0.58 pounds lighter at birth than babies born to smoker mothers.
    2. We are 95% confident that the difference in average weights of babies whose moms are smokers and nonsmokers is between 0.05 to 0.58 pounds.
    3. We are 95% confident that the difference in average weights of babies in this sample whose moms are smokers and nonsmokers is between 0.05 to 0.58 pounds.
    4. We are 95% confident that babies born to nonsmoker mothers are on average 0.05 to 0.58 pounds heavier at birth than babies born to smoker mothers.
      5. from code below, the result shows that the conclusion of the hypothesis test is we are 95% confident that babies born to nonsmoker mothers are on average 0.05 to 0.58 pounds heavier at birth than babies born to smoker mothers. 
# type your code for the Question 5 here, and Knit
inference(y = weight, x = habit, data = nc, statistic = "mean", type = "ci", conf_level = 0.95, method = "theoretical")
## Response variable: numerical, Explanatory variable: categorical (2 levels)
## n_nonsmoker = 873, y_bar_nonsmoker = 7.1443, s_nonsmoker = 1.5187
## n_smoker = 126, y_bar_smoker = 6.8287, s_smoker = 1.3862
## 95% CI (nonsmoker - smoker): (0.0508 , 0.5803)

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 = weight, x = habit, data = nc, statistic = "mean", type = "ci", 
          method = "theoretical", order = c("smoker","nonsmoker"))
## Response variable: numerical, Explanatory variable: categorical (2 levels)
## n_smoker = 126, y_bar_smoker = 6.8287, s_smoker = 1.3862
## n_nonsmoker = 873, y_bar_nonsmoker = 7.1443, s_nonsmoker = 1.5187
## 95% CI (smoker - nonsmoker): (-0.5803 , -0.0508)

  1. Calculate a 99% confidence interval for the average length of pregnancies (weeks). 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. Which of the following is the correct interpretation of this interval?
    1. (38.1526 , 38.5168)
    2. (38.0892 , 38.5661)
    3. (6.9779 , 7.2241)
    4. (38.0952 , 38.5742)
      5. A 99% confidence interval for the average length of pregnancies is (38.0952 , 38.5742) 
# type your code for Question 6 here, and Knit
inference(y = weeks, data = nc, statistic = "mean", type = "ci", conf_level = 0.99, method = "theoretical")
## Single numerical variable
## n = 998, y-bar = 38.3347, s = 2.9316
## 99% CI: (38.0952 , 38.5742)

Exercise: Calculate a new confidence interval for the same parameter at the 90% confidence level. Comment on the width of this interval versus the one obtained in the the previous exercise.

# type your code for the Exercise here, and Knit
inference(y = weeks, data = nc, statistic = "mean", type = "ci",
          conf_level = 0.90, method = "theoretical")
## Single numerical variable
## n = 998, y-bar = 38.3347, s = 2.9316
## 90% CI: (38.1819 , 38.4874)

the result shows that, the 99% and 90% confidence interval for the average length of pregnancies is (38.0952 , 38.5742) for 99% CI and (38.1819 , 38.4874) for 90% CI respectively. As we expect the range of 90% CI is narrower than the range of 99% CI

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

# type your code for the Exercise here, and Knit
inference(y = gained, x = mature, data = nc, statistic = "mean",
          type = "ht", null = 0, alternative = "twosided",
          method = "theoretical")
## Response variable: numerical
## Explanatory variable: categorical (2 levels) 
## n_mature mom = 129, y_bar_mature mom = 28.7907, s_mature mom = 13.4824
## n_younger mom = 844, y_bar_younger mom = 30.5604, s_younger mom = 14.3469
## H0: mu_mature mom =  mu_younger mom
## HA: mu_mature mom != mu_younger mom
## t = -1.3765, df = 128
## p_value = 0.1711

  1. 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.
# type your code for Question 7 here, and Knit
ggplot(nc, aes(x = mature, y = mage)) +
    geom_boxplot()

nc %>% 
    group_by(mature) %>%
    summarise(max_age = max(mage),
              min_age = min(mage))
## `summarise()` ungrouping output (override with `.groups` argument)
## # A tibble: 2 x 3
##   mature      max_age min_age
##   <fct>         <int>   <int>
## 1 mature mom       50      35
## 2 younger mom      34      13
The age cutoff for younger and mature mothers is determined by data manipulation method. Firstly, the results are showed only mature mom and younger mom categories by group_by function and then they are summarized the cutoff age by summarise function to show min and max value. So the maximum age of younger mom is 34 and the minimum age of mature mom is 35.

Exercise: Pick a pair of variables: one numerical (response) and one categorical (explanatory). 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. Be sure to check all assumptions,state your \(\alpha\) level, and conclude in context. (Note: Picking your own variables, coming up with a research question, and analyzing the data to answer this question is basically what you’ll need to do for your project as well.)

the response variable: 'weeks' (length of pregnancy in weeks)
the explanatory variable: 'lowbirthweight'(whether baby was classified as low birthweight (`low`) or not (`not low`).) are selected
ggplot(data = nc, aes(x = lowbirthweight, y = weeks)) +
    geom_boxplot(aes(fill = lowbirthweight)) +
    ggtitle("The relation between Length of pregnancy(weeks) and
            the birth weight") +
    ylab("Length of pregnancy(weeks)") + 
    xlab("Birth weight")
## Warning: Removed 2 rows containing non-finite values (stat_boxplot).

Therefore, we work on hypothesis testing as follows: \(H_0: \mu_{lowbirthWt} = \mu_{notLowbirthWt}\); \(H_A: \mu_{lowbirthWt} \ne \mu_{notLowbirthWt}\)

# type your code for the Exercise here, and Knit
inference(y = weeks, x = lowbirthweight, data = nc, 
          statistic = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical"
          )
## Response variable: numerical
## Explanatory variable: categorical (2 levels) 
## n_low = 110, y_bar_low = 33.4273, s_low = 4.6991
## n_not low = 888, y_bar_not low = 38.9426, s_not low = 1.8947
## H0: mu_low =  mu_not low
## HA: mu_low != mu_not low
## t = -12.1876, df = 109
## p_value = < 0.0001

as the result, The P-value is quite pretty small(<0.0001). Therefore The null hypothesis is rejected anyway. in short, the conclusion is the babies corresponding to not low birth weight are in, on average, longer pregnancy duration than babies with low birth weight.

This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was written for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel.