---
title: 'Inference for numerical data'
output:
  html_document:
    css: ../lab.css
    highlight: pygments
    theme: cerulean
    toc: true
    toc_float: true
---

```{r global_options, include=FALSE}
knitr::opts_chunk$set(eval = FALSE)
library(dplyr)
library(ggplot2)
library(oilabs)
```

## 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 OpenIntro labs, `oilabs`.

Let's load the packages.

```{r load-packages, message=FALSE}
library(dplyr)
library(ggplot2)
library(oilabs)
```

### Creating a reproducible lab report

To create your new lab report, start by opening a new R Markdown document... From Template... then select Lab Report from the `oilabs` package. 

### 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.

```{r load-data}
data(nc)
```

We have observations on 13 different variables, some categorical and some 
numerical. The meaning of each variable can be found by bringing up the help file:

```{r help-nc}
?nc
```


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

Remember that you can answer this question by viewing the data in the data viewer or
by using the following command:

```{r str}
glimpse(nc)
```

## 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 `favstats` function from `mosaic` can be useful.

```{r summary}
library(mosaic)
favstats(~gained, data = nc)
```

1.  How many mothers are we missing weight gain data from?

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.

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

```{r by-means}
nc %>%
  group_by(habit) %>%
  summarise(mean_weight = mean(weight))
```

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

## Inference

3.  Are all conditions necessary for inference satisfied? Comment on each. You can 
compute the group sizes with the `summarize` command above by defining a new variable
with the definition `n()`.

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

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

```{r inf-weight-habit-ht, tidy=FALSE}
inference(y = weight, x = habit, data = nc, statistic = "mean", type = "ht", null = 0, 
          alternative = "twosided", method = "theoretical")
```

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'll 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`.

5.  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.

By default the function reports an interval for ($\mu_{nonsmoker} - \mu_{smoker}$)
. We can easily change this order by using the `order` argument:

```{r inf-weight-habit-ci, tidy=FALSE}
inference(y = weight, x = habit, data = nc, statistic = "mean", type = "ci", 
          method = "theoretical", order = c("smoker","nonsmoker"))
```

* * *

## More Practice

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.

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: `conf_level = 0.90`. Comment on the width of this interval versus 
the one obtained in the previous 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.

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.

10. 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.)

<div id="license">
This is a product of OpenIntro that is released under a [Creative Commons 
Attribution-ShareAlike 3.0 Unported](http://creativecommons.org/licenses/by-sa/3.0).
This lab was adapted for OpenIntro by Mine &Ccedil;etinkaya-Rundel from a lab 
written by the faculty and TAs of UCLA Statistics.
</div>