09-STAT_220_TwoSamplePreLab-formula.Rmd

---
title: 'Inference for two samples pre-lab: formula'
author: "Professor McNamara"
output:
  html_document:
    toc: true
    toc_depth: 3
    toc_float: true
---

## Setup and packages

We use three packages in this course: `Lock5Data`, `mosaic` and `ggformula`. To load a package, you use the `library()` function, wrapped around the name of a package. I've put the code to load one package into the chunk below. Add the other two you need. 

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, error = TRUE)
library(Lock5Data)
# put in the other two packages you need here
```

## Loading in data

As usual, we'll load the example data, `GSS_clean.csv`. It should be inside the `data` folder in your RStudio Cloud. We'll use the `read.csv()` function to read in the data. 

```{r data-load}
GSS <- read.csv("data/GSS_clean.csv")
```

We also need to do a little data cleaning to ensure this will work properly for the lab,

```{r}
GSS <- filter(GSS, should_marijuana_be_made_legal != "")
GSS <- filter(GSS, self_emp_or_works_for_somebody != "")
```

## Inference for two samples

This lab talks about "inference for two samples"-- in other words, doing inference on two proportions, or on two means. Like usual, we'll be using distributional approximations (either the normal distribution, or the t-distribution, depending on the parameter we're studying), and performing our main inferential tasks (confidence intervals and hypothesis tests). 

## Difference of proportions

To do inference for a difference of proportions, we approximate using a normal distribution. We need a way to compute the standard error, and the formula is different based on whether we're doing a confidence interval or a hypothesis test.

For two proportions, in a confidence interval
$$
SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}
$$

For two proportion, in a hypothesis test

$$
SE=\sqrt{\frac{\hat{p}(1-\hat{p})}{n_1}+\frac{\hat{p}(1-\hat{p})}{n_2}}
$$
Where $\hat{p}$ is the pooled proportion. 

Let's try making a confidence interval for a difference of two proportions. I'd like to find a confidence interval for the difference of proportion for people who said marijuana should be made legal, based on whether the person is self employed or works for somebody else.

As with last time, you can either use R as a calculator or use the shortcut functions. Let's focus on the functions. 

Let's start by finding our observed statistic. 

```{r}

```


Now, let's do a 90% confidence interval. What needs to be changed here in order to make it a 90% interval?

```{r}
prop.test(should_marijuana_be_made_legal ~ self_emp_or_works_for_somebody, data = GSS)
```

How can we interpret the interval? 



We could also do a hypothesis test. What are the hypotheses we're testing?


In order to do the test, we just add a few more arguments to our function. 
```{r}
prop.test(should_marijuana_be_made_legal ~ self_emp_or_works_for_somebody, data = GSS)
```

## Inference for a difference of means

To do inference for a difference of means, we approximate using a t-distribution. We need a way to compute the standard error, but the formula is not different between confidence intervals and hypothesis tests. 
$$
SE = \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}
$$
Again, we can either use R as a big calculator or we can use functions. 

I'm interested in seeing whether the mean number of years of school completed is different between people born in the US and those born outside the US. 

We'll start by computing the point estimate,

```{r}

```


Now let's make a confidence interval for that difference of means,

```{r}
t.test()
```

How can we interpret this interval?



Now, let's perform a hypothesis test. We want to know if the means are significantly different from one another. How would we write out our hypotheses?

In order to perform the test, we add some more parameters,

```{r}

```

What do we conclude at the 5% level? The 10% level?

## Inference for paired data

Finally, we want to think about paired data. There are many ways to do this in R, but the easiest is to just make a new variable of differences, and then do inference on that variable. 

Because a person's mother and father are "naturally paired," if we want to see if the mean level of education is different between mothers and fathers, we should really make a new variable of the differences and do inference on that. 

```{r}
GSS <- transform(GSS, diff = highest_year_school_completed_father - highest_year_school_completed_mother)
```

Now, we can do inference just as we would for a single mean, starting with finding the point estimate,

```{r}

```

Now, we can do inference just the way we did in the single sample lab. 

Here is the code from the last lab. Can you transform it to work on our differences data?

```{r}
t.test(~number_of_hours_worked_last_week, data = GSS, alternative = "two.sided", mu = 40)
```

What is the conclusion to our test?