NormalCurveAreaBetween.Rmd

---
title: "Normal Area Between"
params:
  a: 3.3 
  b: 5.5 
  mu: 5 
  sigma: 1.2 
jupyter:
  kernelspec:
    display_name: R
    language: R
    name: ir
---

```{r options, eval=FALSE, include=FALSE}
options(repr.plot.width=4, repr.plot.height=2.5)
```

```{r params, include=FALSE}
if(!exists("params")) {
    params<-list(a=3.3,b=5.5,mu=5,sigma=1.2)
}
a<-params$a 
b<-params$b 
mu<-params$mu 
sigma<-params$sigma 
```

```{r computed, include=FALSE}
lowerz<-round((a-mu)/sigma,2)
upperz<-round((b-mu)/sigma,2)
areaB <- round(pnorm(upperz),4)
areaA <- round(pnorm(lowerz),4)
areaBetween <- round(areaB-areaA,4)
```

```{r graphicsutils, include=FALSE}
source("graphics_utils.R")
```

We are going to find the area between the two x-values $x=`r a`$ and $x=`r b`$ for a normal distribution with mean of $\mu = `r mu`$ and with a standard deviation of $\sigma = `r sigma`$.   

Here is what the area looks like: 

```{r xarea, include=TRUE, echo=FALSE, fig.align="center", fig.width=4, fig.height=2.5} 
normal_draw_area_between(a,b,mu,sigma)
```

We will convert both the x-values to z-values, then find the area between those z-values using the standard normal distribution (z distribution).

Lets find the "lower z-value" first. This is the z-value for the left endpoint ($x=`r a`$):

$$
z =\frac{x-\mu}{\sigma} =\frac{`r a`-`r mu`}{`r sigma`} =\frac{`r a-mu`}{`r sigma`} =`r round(lowerz,2)`
$$

Next lets find the "upper z-value". This is the z-value for the right endpoint ($x=`r b`$):

$$
z =\frac{x-\mu}{\sigma} =\frac{`r b`-`r mu`}{`r sigma`} =\frac{`r b-mu`}{`r sigma`} =`r round(upperz,2)`
$$

Now the area that we want that is between $z = `r lowerz`$ and $z = `r upperz`$

```{r zarea, include=TRUE, echo=FALSE, fig.align="center", fig.width=4, fig.height=2.5} 
z_draw_area_between(lowerz,upperz)
```

We will find the area between these two z-values by subtracting the left tail areas for each one.       

Using the standard normal table we find the left tail area for $z=`r upperz`$ first:

$$
\text{left tail area for `r upperz`} =`r areaB`
$$

Then we find the left tail area for $z = `r lowerz`$ next: 

$$
\text{left tail area for `r lowerz`} =`r areaA`
$$

Finally subtract these two areas to find the area between the two: 

$$
\text{area between}=`r areaB` - `r areaA`=`r areaBetween`
$$