Lecture 4

class: center, middle, inverse, title-slide

# Lecture 4
### Jung-Jin Lee
### Jan 29, 2019

---

## Normal distribution

Consider the data set `heights.txt` which consists of mother/daughter height pairs.

```r
ht <- read.table(file = "heights.txt", header = T, sep = " ")
dim(ht)
```

```
## [1] 1375    2
```

```r
head(ht)
```

```
##   Mheight Dheight
## 1    59.7    55.1
## 2    58.2    56.5
## 3    60.6    56.0
## 4    60.7    56.8
## 5    61.8    56.0
## 6    55.5    57.9
```

---

## Normal distribution, continued

.pull-left[

```r
ggplot(ht, aes(Mheight)) + 
  geom_histogram()
```

<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" />
]

.pull-right[

```r
ggplot(ht, aes(Dheight)) + 
  geom_histogram()
```

<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-4-1.png" style="display: block; margin: auto;" />
]

Note that the histograms look like a bell.

---

## Normal distribution, continued

Bell shapes are observed in other data sets, too.

.pull-left[

```r
ggplot(diamonds, aes(depth)) + 
  geom_histogram()
```

<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-5-1.png" style="display: block; margin: auto;" />
]

.pull-right[

```r
ggplot(iris, aes(Sepal.Width)) + 
  geom_histogram() 
```

<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-6-1.png" style="display: block; margin: auto;" />
]

---

## Rigorous definition of normal distribution

- A continuous random variable `$X$` is said to follow a _normal distribution_ `$N(\mu, \sigma^2)$` if the probability `$P(a\leq X \leq b)$` that `$X$` is between `$a$` and `$b$` is given by 
`$$P(a\leq X \leq b) = \int_a^b \phi_{\mu,\sigma}(x)\,dx,$$`
where `$$\phi_{\mu,\sigma}(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}.$$`

- In particular, the area under the curve equals 1.

- `$\phi_{\mu,\sigma}(x)$` is a bell-shaped function called the probability distribution function (pdf) of `$X$`. `$\mu$` is called the _mean_ and `$\sigma$` is called the _standard deviation_.

- In particular, if `$\mu = 0$` and `$\sigma = 1$`, `$X$` is said to follow the _standard_ normal distribution.

---

## Why is normal distribution important?

- Has mathematically important properties (e.g. Central Limit Theorem).

- Related with other important distributions (e.g. `$t$`, `$F$`, `$\chi^2$`).

- Provides a good model for many real-world data (e.g. A normal distribution is a good model for data such as heights or weights).

---

## Normal pdf

.pull-left[
<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-8-1.png" style="display: block; margin: auto;" />
]

.pull-right[
<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-9-1.png" style="display: block; margin: auto;" />
]

_mean_ determines the center of the bell shape, while _standard deviation_ measures the spread.

---

## Normal pdf, continued

**Example**: the life of a fully-charged cell phone battery is normally distributed with a mean of 13 hours with a standard deviation of 1 hour. `$X$`, the life of a randomly selected fully-charged phone battery, can be assumed to follow `$N(13, 1^2)$`.

---

## Normal pdf, continued

- What is the probability that `$X \leq 13$`?

- As the curve is symmetric, the desired probability is `$0.5$`.

---

## Normal pdf, continued

- What is the probability that `$X \leq 12$`?

.pull-left[
<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-12-1.png" style="display: block; margin: auto;" />
]

.pull-right[
`pnorm()` computes the desired probability:

```r
pnorm(12, mean = 13, sd = 1)
```

```
## [1] 0.1586553
```

```r
pnorm(13, mean = 13, sd = 1)
```

```
## [1] 0.5
```
]

**Exercise**: compute the probability that X `$\geq$` 14.

---

## Normal pdf, continued

- What is the probability that `$11 \leq X \leq 14$`?

.pull-left[
<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-14-1.png" style="display: block; margin: auto;" />
]

.pull-right[
<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-15-1.png" style="display: block; margin: auto;" />
]

---

## Normal pdf, continued

**Exercise**:

- Compute the probability that `$X\geq 15$`, where `$X$` follows `$N(13,1^2)$`.
- Compute the probability that `$Z \leq -1$`, where `$Z$` follows `$N(0,1^2)$`.
- Determine `$z_{0.1}$` such that the probability that `$Z\leq z_{0.1}$` equals 0.9 (_hint_: look up `qnorm()`).

.pull-left[
<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-16-1.png" width="90%" style="display: block; margin: auto;" />
]

.pull-right[
<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-17-1.png" width="90%" style="display: block; margin: auto;" />
]

---

## _t_-distribution

- Class of bell-shaped distribution determined by a parameter called _degrees of freedom_.

---

## _t_-distribution, continued

**Example**: suppose `$X$` follows `$t$`-distribution with `$5$` degrees of freedom. What is the probability that `$X\leq 0$`? `$X\leq -1.5$`? `$-1 \leq X \leq 1.2$`? `$X \geq 1.6$`?

.pull-left[
<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-19-1.png" style="display: block; margin: auto;" />
]

.pull-right[

```r
pt(0, df = 5)
```

```
## [1] 0.5
```

```r
pt(-1.5, df = 5)
```

```
## [1] 0.09695184
```

```r
pt(1.2, df = 5) - pt(-1, df = 5)
```

```
## [1] 0.6764457
```

```r
1-pt(1.6, df = 5)
```

```
## [1] 0.08524762
```
]

---

## _t_-distribution, continued

**Exercise**: suppose `$X$` follows `$t$`-distribution with `$5$` degrees of freedom. Determine `$t_{5, 0.2}$` so that the probability that `$X\geq t_{5, 0.2} = 0.2$` (_Hint_: look up `qt()`).

---

## _t_-distribution, continued

**Exercise**: suppose `$X$` follows `$t$`-distribution with `$4$` degrees of freedom. Determine `$t_{4, 0.025}$` so that the probability that `$-t_{4, 0.025} \leq X \leq t_{4, 0.025} = 0.95$`.

---

## Using data to make inference

- Point estimate

- Confidence interval

- Statistical tests

---

## Point estimate

- Assume that the heights of all UK women follow a normal distribution `$N(\mu, \sigma^2)$`.
- Interested in determining `$\mu$` and `$\sigma$`.
- Sample mean ans sample standard deviation are known to be _good_ estimates for `$\mu$` and `$\sigma$`, respectively.

.pull-left[
<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-23-1.png" width="95%" style="display: block; margin: auto;" />
]

.pull-right[

```r
mean(ht$Mheight)
```

```
## [1] 62.4528
```

```r
sd(ht$Mheight)
```

```
## [1] 2.355103
```

**Exercise**: compute the sample mean and sample standard deviation of `Sepal.Width` in `iris` data set.

]

---

## Confidence interval

Point estimate is too naive -- how much can we trust 62.4528 as a reliable estimator for `$\mu$`, the true population mean?

**Example**: suppose that we are measuring the battery life which is assume to follow a normal distribution. Assume that we collected two sets of samples of size 5:
`$$\text{Sample 1: }\{12, 13, 13, 14, 13\}$$`
`$$\text{Sample 2: }\{10, 11, 16, 13, 15\}$$`

```r
sample1 <- c(12, 13, 13, 14, 13); sample2 <- c(10, 11, 16, 13, 15)
c(mean(sample1), mean(sample2))
```

```
## [1] 13 13
```

```r
c(sd(sample1), sd(sample2))
```

```
## [1] 0.7071068 2.5495098
```

---

## Confidence interval, continued

Suppose that a sample of size `$n$`, say `$X_1, X_2, \ldots, X_n$`, is taken from a population that follows `$N(\mu, \sigma^2)$`. The 100 `$\times (1-\alpha)$` % confidence interval of `$\mu$` is defined as
`$$\left(\bar{X} - t_{n-1, \alpha/2}\frac{S}{\sqrt{n}},~\bar{X} + t_{n-1, \alpha/2}\frac{S}{\sqrt{n}}\right),$$`
where, `$\bar{X}$` and `$S$` represent the sample mean and sample standard deviation, respectively and `$t_{n-1, \alpha/2}$` is the quantile value of a `$t$`-distribution of `$n-1$` degrees of freedom with tail probability `$\alpha/2$`.

In particular, the 95% confidence interval (so `$\alpha = 0.05$`) of `$\mu$` is given by 
`$$\left(\bar{X} - t_{n-1, 0.025}\frac{S}{\sqrt{n}},~\bar{X} + t_{n-1, 0.025}\frac{S}{\sqrt{n}}\right),$$`

Interpretation of 95% confidence interval: when this procedure of creating 95% confidence intervals is repeated, about 95% of the intervals will contain the true population mean.

---

## Confidence interval, continued

```r
alpha <- 0.05
xbar <- mean(sample1)
sdev <- sd(sample1)
n <- length(sample1)
tval <- qt(1-alpha/2, df = n-1)
CI95 <- xbar + tval*sdev/sqrt(n)*c(-1,1)
CI95
```

```
## [1] 12.12201 13.87799
```

**Exercise**: compute the 95% confidence interval of `sample2`.

---

## Confidence interval, continued

`t.test()` can be used to compute confidence interval of mean.

```r
t1 <- t.test(sample1)
t1
```

```
## 
## 	One Sample t-test
## 
## data:  sample1
## t = 41.11, df = 4, p-value = 2.093e-06
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  12.12201 13.87799
## sample estimates:
## mean of x 
##        13
```

---

## Confidence interval, continued

```r
t2 <- t.test(sample2)
t2
```

```
## 
## 	One Sample t-test
## 
## data:  sample2
## t = 11.402, df = 4, p-value = 0.0003375
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##   9.834366 16.165634
## sample estimates:
## mean of x 
##        13
```

---

## Confidence interval, revisited

Review of confidence interval:

`$$\left(\bar{X} - t_{n-1, \alpha/2}\frac{S}{\sqrt{n}},~\bar{X} + t_{n-1, \alpha/2}\frac{S}{\sqrt{n}}\right)$$`

- As `$1-\alpha$` (called confidence level) increases, the confidence interval becomes wider.

- As `$n$` (sample size) increases, the confidence interval becomes narrower.

---

## Exercises

**Exercise**: based on the `heights.txt` data, find the 80%, 90% and 95% confidence interval of the mean of UK women's heights (mom's heights).

**Exercise**: `starwars`, which comes with the package `dplyr`, is a data set that contains data related with characters in the movie Star Wars.

- Explorer the data set. Subset the data so that the resulting data set contains only human species with `height` data available (_Hint_: look up `is.na()`). How many observations?

---

## Exercises, continued

- Explore the data set and create the following plot showing the height distribution of Human species:
<img src="Drexel_2019_Lecture_4_files/figure-html/unnamed-chunk-29-1.png" width="50%" style="display: block; margin: auto;" />

- Find the name of the tallest Human female character.

- Find the 95% confidence interval of the mean of Human male heights.