Statistical Inference

# Statistical Inference
### Bora Jin

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## Today's Goal

- Understand statistical process terminology
- Understand different types of conclusions we can make through statistical inference
- Understand point estimate and confidence intervals

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# Statistical Inference

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## Terminology

- **Population**: a group of individuals or objects we are interested in studying

- **Parameter**: a numerical quantity derived from the population
(almost always unknown)

- Parameters could be the mean, median, correlation, maximum, etc.

If we had data from every unit in the population, we could just calculate 
population parameters and be done!

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## Terminology

Unfortunately, we usually cannot do this, so we draw conclusions from

- **Sample**: a subset of our population of interest

- **Statistic**: a numerical quantity derived from a sample

- Statistics could be the mean, median, correlation, maximum, etc.

Naturally, it makes sense to use the sample mean (and other quantities derived from the sample) to make generalizations about the population mean.

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## Statistical Inference

Statistical inference is the process of using sample data to make 
conclusions about the underlying population the sample came from.

- **Estimation**: using the sample to estimate a plausible range of values 
for the unknown parameter

- **Testing**: evaluating whether our observed sample provides evidence 
for or against some claim about the population

Today we will focus on **Estimation**.

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# Estimation

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## Trip to Asheville!

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## Asheville Airbnb

- **Research question**: What is the mean price per guest per night
among Airbnb rentals with at least ten reviews in Asheville (zip codes 28801 - 28806)?

- **Population of interest**: listings in Asheville with at least ten reviews.

- **Parameter of interest**: mean price per guest per night among these 
listings.

We have data on the price per guest (`ppg`) for a random sample of 50 Airbnb listings
with at least ten reviews in Asheville, NC, that were active on June 25, 2020 (Source: http://insideairbnb.com/).

- **Sample**: randomly selected 50 listings in the Asheville with at least ten reviews.

- **Statistic**: mean price per guest per night among these sampled listings.

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## Point Estimate

A **point estimate** is a single value computed from the sample data to serve
as the "best guess", or estimate, for the population parameter.

```r
abb <- read_csv("data/asheville.csv")
abb %>% 
  summarize(mean_price = mean(ppg))
```

```
## # A tibble: 1 × 1
##   mean_price
##        <dbl>
## 1       76.6
```

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## Visualizing Our Sample

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## Point vs. Interval

If you want to catch a fish, do you prefer a spear or a net?

.pull-left[
<img src="img/13/spear.png" width="300" style="display: block; margin: auto;" />
]
.pull-right[
<img src="img/13/net.png" width="300" style="display: block; margin: auto;" />
]

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## Point vs. Interval

If you want to estimate a population parameter, do you prefer to report a single value or a range of values the parameter might be in?

.pull-left[
<img src="img/13/spear.png" width="300" style="display: block; margin: auto;" />
]
.pull-right[
<img src="img/13/net.png" width="300" style="display: block; margin: auto;" />
]

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## Point vs. Interval

If you want to estimate a population parameter, do you prefer to report a single value or a range of values the parameter might be in?

.pull-left[
<img src="img/13/spear.png" width="300" style="display: block; margin: auto;" />
]
.pull-right[
<img src="img/13/net.png" width="300" style="display: block; margin: auto;" />
]

- If we report a point estimate, we probably won’t hit the exact population 
parameter.

- If we report a range of plausible values, we have a good shot at capturing 
the parameter.

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# Uncertainty Quantification

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## Confidence Intervals

- A plausible range of values for the population parameter is a **confidence interval**.

- In order to construct a confidence interval we need to quantify variability (uncertainty) of our sample statistic.

- For example, if we want to construct a confidence interval for a population mean, we need to come up with a plausible range of values around our observed sample mean.

- This range will depend on how precise and how accurate our sample mean is as an estimate of the population mean.

- Quantifying this requires a measurement of how much we would expect the sample mean to vary from sample to sample.

---

## Quantifying Variability

- There is almost always some variability of sample statistics because random samples may differ from each other.

- If we took another random sample of 50 Airbnb listings in Asheville, we probably wouldn't get the same mean price per guest.

- Suppose we split the class in half and ask each student their height. 
Then, we calculate the mean height of students on each side of the classroom. 
Would you expect these two means to be exactly equal, close but not equal, 
or wildly different?

- Suppose you randomly sample 50 students and 5 of them are left handed. If you 
were to take another random sample of 50 students, how many would you expect to 
be left handed? Would you be surprised if only 3 of them were left handed? Would 
you be surprised if 40 of them were left handed?

---

## Quantifying Variability

We can quantify the variability of sample statistics using different approaches:

- **Simulation**: via bootstrapping or "resampling" techniques

- **Theory**: via the Central Limit Theorem

Today we will focus on **Booststrapping**.

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# Bootstrapping

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## Bootstrapping

- The term **bootstrapping** comes from the phrase "pulling oneself up by one’s 
bootstraps", which is a metaphor for accomplishing an impossible task without 
any outside help.

- **Impossible task**: Estimating a population parameter using data from only the given sample

- **Note**: This notion of saying something about a population parameter using 
only information from an observed sample is the crux of statistical inference,  
not limited to bootstrapping.

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## Bootstrappng Steps

1. Take a bootstrap sample: a random sample taken **with replacement** from the original sample, 
**of the same size** as the original sample.

2. Calculate the bootstrap statistic: the statistic you’re interested in (the 
mean, the median, the correlation, etc.) computed on the bootstrap sample.

3. Repeat steps (1) and (2) many times to create a bootstrap distribution: 
a distribution of bootstrap statistics.

4. Calculate the bounds of the XX% confidence interval as the middle XX% of 
the bootstrap distribution.

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## Original Sample

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## Step 1

Take a bootstrap sample: a random sample taken **with replacement** from the original sample, 
**of the same size** as the original sample.

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## Step 2

Calculate the bootstrap statistic (in this case, the sample mean) using the bootstrap sample.

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## Step 3

Do steps 1 and 2 over and over again to create a bootstrap distribution of sample means.

<img src="13-inference2_BJ_files/figure-html/unnamed-chunk-16-1.png" width="60%" height="90%" style="display: block; margin: auto;" />
              
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## Step 3

In this plot, we've taken 1,000 bootstrap samples, calculated the
sample mean for each, and plotted them in a histogram.

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## Step 3

Here we compare the bootstrap distribution of sample means to that 
of the original data. What do you notice?

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## Step 4

Calculate the bounds of the bootstrap interval by using percentiles of the bootstrap distribution.

<img src="13-inference2_BJ_files/figure-html/unnamed-chunk-19-1.png" width="50%" style="display: block; margin: auto;" />
--

The 95% confidence interval for the mean price per guest per night
among Airbnb rentals with at least ten reviews in Asheville is ($64, $90).

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# Questions?

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## Bulletin

- Watch videos for [Prepare: June 1](https://sta199-summer22.netlify.app/prepare/week04_jun01_BJ.html)

- Mid-course feedback on [Sakai](https://sakai.duke.edu/x/YNKidU) due Fridy, June 3 at 11:59pm

- Kick-off of the final project! Read [intructions](https://sta199-summer22.netlify.app/project/project_BJ.html) carefully

- Project proposal due Friday, June 3 at 11:59pm

- Lab 04 due tonight at 11:59pm