Central Limit Theorem

# Central Limit Theorem
### Dr. Dogucu

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<div class="my-footer"> 
 Copyright &copy; <a href="https://mdogucu.ics.uci.edu">Dr. Mine Dogucu</a>. <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a></div>

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## Data

We will be using payroll data from Los Angeles Police Department (LAPD) from 2018.

```r
glimpse(lapd)
```

```
## Rows: 14,824
## Columns: 1
## $ base_pay <dbl> 119321.60, 113270.70, 148116.00, 78676.87, 109373.63, 9500...
```

---

## Population Distribution

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class: middle

## Population Mean

We have data on everyone who worked for LAPD in the year 2018. So the distribution we just looked at is a population distribution. We can go ahead and calculate the population mean ( `$\mu$` ).

```r
summarize(lapd, mean(base_pay))
```

```
## # A tibble: 1 x 1
##   `mean(base_pay)`
##              <dbl>
## 1           85149.
```

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## Population Standard Deviation

We can calculate the population standard deviation ( `$\sigma$` ).

```
## # A tibble: 1 x 1
##   `sd(base_pay)`
##            <dbl>
## 1         38423.
```

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What if we did not have access to all this data? What would we do?

Rely on a sample!

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Let's assume we went ahead and took a (random) sample of LAPD staff and asked their salary information (and they report to us truthfully) and calculated a mean, would we find a mean of 85149.05? Why, why not?

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Let's pretend we have never seen the data and we do not know the population parameter `$\mu$`. In fact this is usually what happens in real life. We do not have the population information but we do want to know a population __parameter__ (does not necessarily have to be the mean).

If we took a sample and calculated the sample mean, we would name this __point estimate__ of the parameter.

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class: middle center

|                               | Parameter of Interest | Point Estimate / Sample Statistic |
|-------------------------------|-----------------------|-----------------------------------|
| Mean                          | `$\mu$`                 | `$\bar x$`                          |
| Difference of Two Means       | `$\mu_1 - \mu_2$`       | `$\bar x_1 - \bar x_2$`             |
| Proportion                    | `$\pi$`                 | `$p$`                               |
| Difference of Two Proportions | `$\pi_1 - \pi_2$`       | `$p_1 - p_2$`                       |

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class: middle

## First Sample

We would like to know about `$\mu$` but we cannot access the whole population.

A researcher takes a random sample of 20 LAPD staff and ask them about their base pay.

```
##  [1]      0.00 109368.20  95924.46  29417.88  32236.80  98306.29      0.00
##  [8]  95877.27      0.00  61521.20 109054.97  53726.44  89835.29      0.00
## [15] 109378.40  69640.00  43810.12 109409.10 103408.00   3600.00
```

**Mean of first sample**, `$\bar x_1$` =

```
## [1] 60725.72
```

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## Mean of second sample

`$\bar x_2$` =

```
## [1] 81837.23
```

## Mean of third sample

`$\bar x_3$` =

```
## [1] 85614.37
```

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We could do this over and over again. Don't you worry! I did it.

I have taken 10,000 samples of size 200 (sample size of 20 is just too small) and calculated their mean. The following slide shows the distribution of the **sample means**.

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### Sampling Distribution of the Mean

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### Sampling Distribution of the Mean

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## Conclusion

When certain conditions are met then:

`$$\bar x \sim \text{approximately }N( \mu, \frac{\sigma^2}{n})$$`

`$$(\bar x_1 - \bar x_2) \sim \text{approximately } N(\mu_1 - \mu_2, \frac{\sigma_1^2}{n_1}+ \frac{\sigma_2^2}{n_2})$$`

`$$p \sim  \text{approximately } N(\pi,  \frac{\pi(1-\pi)}{n})$$`

`$$(p_1 - p_2) \sim \text{approximately } N((\pi_1 - \pi_2),  {\frac{\pi_1(1-\pi_1)}{n_1} + \frac{\pi_2(1-\pi_2)}{n_2}})$$`

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## Central Limit Theorem (CLT)

If certain conditions are met, the sampling distribution will be normally distributed with a mean equal to the population parameter. The standard deviation will be inversely proportional to the square root of the sample size.

We will learn the conditions in the upcoming lectures.

Moving forward we will use CLT to make __inference__ about population parameters using sample statistics.