The third homework assignment is designed to give you with some practice with both the T-test as well as the basic one way ANOVA model. I have divided this homework into two separate parts. In the first part, I will give questions to myself and then proceed to answer them and show you the code along the way. In the second exercise I will let you have the fun and task you with providing the answers to various questions along the way. In the part of the exercise that you do, you have to explain the appropriate statistical analysis along with appropriate R code and figures and tables. Don't just put a fact and figure down without decent explanation.

\textcolor{blue}{My pt(),qt(),pchisq(), qchisq(), pf(),and qf() Problems}

The $T$-distribution, the $\chi^2$ distribution (denoted chi-squared) and the $F$-distribution play an important role in statistics as they are involved in hypothesis testing for all kinds of problems.

The T-distribution is completely characterized by the degrees of freedom. If $v$ is the degrees of freedom then the PDF of this distribution looks like a bell curve centered at zero with wider tails. For $v$ small, the bell curve is a lot wider than the normal distribution. As $v$ gets large, the $T$ distribution with $v$ degrees of freedom approaches a normal distribution. Shown below are the PDF functions for the $T$ distribution with $5 (red),10 (green),20 (blue)$ and the normal distribution which is a $T$ distribution with $\infty$ degrees of freedom. The following plots show you what the $T$ distribution looks like for various degrees of freedom.

```{r}
curve(dt(x,5),from=-4,to=4,col="red",lwd=3)
curve(dt(x,10),from=-4,to=4,col="dark green",lwd=3,add=TRUE)
curve(dt(x,20),from=-4,to=4,col="blue",lwd=3,add=TRUE)
curve(dnorm(x),from=-4,to=4,col="black",lwd=3,add=TRUE)
```

The chi-square and F distributions are positive distributions. Here is what the $\chi^2_{5}$ distribution looks like
```{r}
df=5
curve(dchisq(x,df),from=0,to=4*df,col="red",lwd=3)
```

Here is what the $F$idistribtion looks like with 5 numerator degrees of freedom and 10 denominator degrees of freedom

```{r}
curve(df(x,5,10),from=0,to=4,col="red",lwd=3)
```

The folowing functions should be helpful when attempting to shade in areas under these distributions

```{r}
shadetdist=function(a,b,df){
curve(dt(x,df),from=-4,to=4)
x=seq(a,b,length=100)
y=dt(x,df);xx=c(a,x,b);yy=c(0,y,0)
polygon(xx,yy,col="pink")
}
```

```{r}
shadechisq=function(a,b,df){
U = qchisq(.999,df)
curve(dchisq(x,df),from=0,to=U)
x=seq(a,b,length=100)
y=dchisq(x,df);xx=c(a,x,b);yy=c(0,y,0)
polygon(xx,yy,col="pink")
}
```

```{r}
shadeFdist=function(a,b,ndf,ddf){
#ndf = numerator df, ddf = denominator df
U = qf(0.999,ndf,ddf)
curve(df(x,ndf,ddf),from=0,to=U)
x=seq(a,b,length=100)
y=df(x,ndf,ddf);xx=c(a,x,b);yy=c(0,y,0)
polygon(xx,yy,col="pink")
}
```

If $R$ is a random variable that follows a $T$ distriburion with $K$ degrees of freedom, the way we say this compactly in math is $R \sim T_{K}$ which means $R$ is disributed as a $T$ distribution with $K$ degrees of freedom.

a. Suppose that $R \sim T_{5}$ find the area under the $T$ distribution bell curve less than 1.2, or $P(R < 1.2)$.

```{r}
shadetdist(-4,1.2,5)
```
```{r}
ans=pt(1.2,5)
ans
```
$$ P(R < 1.2) = pt(1.2,5) \approx 0.858$$

b. If $R \sim T_{10}$, find $P(R > 0.5)$.

```{r}
shadetdist(0.5,4,10)
```
```{r}
ans = 1-pt(0.5,10)
ans
```

$$ P(R > 0.5) = 1- pt(0.5,10) \approx 0.314$$

c. If $R \sim T_{7}$, find the value of $r$ so that $P(R > r) = 0.75$.

\textcolor{red}{Here the problem is asking for the value on the horizontal axis where $0.75$ of the area under the curve is to the right. The answer to that is qt(1-0.75,7). Graphically we can depict this by the following. }

```{r}
r = qt(1-0.75,7)
r
```

```{r}
shadetdist(r,4,7)
text(r,0,"r")
```

If $Z_1, \ldots, z_k$ are $k$ independent $N(0,1)$ random variables then $Q = Z_1^2 + \cdots + Z_k^2$ has a Chi-Squared distribution. In statistics, Chi-squared random variables arise all over the place as this distribution is incredibly important. If $Q$ has a chi-squared distribution with $k$ degrees of freedom we abbreviate this by saying $Q \sim \chi^2_k$. The chi-squared distribution is positive distribution as the $P(Q \leq 0) = 0$. The distribution is skewed to the right, but as k gets large the distribution starts looking like a bell curve. Here is a plot of the chi-squared distribution for $k=5$(red),10 (green),20 (blue) degrees of freedom

d. Suppose that $R \sim \chi^{2}_{10}$ find the area under the $\chi^{2}_{10}$ distribution less than 12, or $P(R < 12)$

\textcolor{red}{The area to the left of $q = 12$ corresponds to $P(R < r)$ which is given by pchisq(12,10) in R. We can graphically
depict this area in R using the following}

```{r}
shadechisq(0,12,10)
```
```{r}
ans=pchisq(12,10)
ans

Suppose you randomly draw 50 values from an unknown distribution....