STAT537/hw7.Rmd at master · runawayhorse001/STAT537 · GitHub

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---
title: "Homework #7"
author: "Ming Chen & Wenqiang Feng"
date: "2/2/2017"
output: html_document
---

## You can also get the [PDF format](./pdf/HW7.pdf) of Wenqiang's Homework.


### Load data

```{R}
riverValley = read.csv("./data/ArkansasRiverValley.csv")
```

### Correlation

* (1). scatterplot

```{R fig.width=5, fig.height=5}
plot(Yield~GrainSize, data=riverValley, xlab="Grain Size", ylab="Yield")
```

* (2). Compute
    + Pearson $r = 0.667871$
    + Spearmans $\rho$ = 0.7634203


```{R}
## Pearson's r
cor(riverValley$GrainSize, riverValley$Yield, method="pearson")

## Spearman's rho
cor(riverValley$GrainSize, riverValley$Yield, method="spearman")
```


* (3). Test hypothesis
    + a. P value = 7.543e-09 < 0.05, reject $H_{0}$. There is a significant correlation between grain size and yield.
    + b. P value = 2.059e-12 < 0.05, reject $H_{0}$. There is a significant correlation between grain size and yield.

```{R warning=F}
cor.test(riverValley$GrainSize, riverValley$Yield, method="pearson")
cor.test(riverValley$GrainSize, riverValley$Yield, method="spearman")
```

* (4). 95% confidence interval for $\rho$ = [0.630624, 0.8527845]

```{R warning=F}
library(mada)
rho = cor.test(riverValley$GrainSize, riverValley$Yield, method="spearman")$estimate
CIrho(rho, dim(riverValley)[1])
```


### Regression


```{R}

```

* (1). Scatterplot

```{R fig.width=5, fig.height=5}
rivervalley.lmfit = lm(Yield~GrainSize, data=riverValley)
plot(Yield~GrainSize, data=riverValley)
abline(rivervalley.lmfit, col="blue", lwd=2)
```

* (2). Estimates
    + $\beta_{0} = -9.294$
    + $\beta_{1} = 744.979$
```{R}
rivervalley.lmfit
```

* (3). Test the hypothesis
    + p value = 7.54e-09 < 0.05, reject $H_{0}$

```{R}
summary(rivervalley.lmfit)
```

* (4) Residuals

```{R}
res = rivervalley.lmfit$residuals
fitted = rivervalley.lmfit$fitted.values
y = riverValley$Yield
data.frame(Y=y, Fitted_value = fitted, Residuals = res)
```