---
title: 'STAT 3202: Practice 09'
author: "Autumn 2018, OSU"
date: ''
output:
html_document:
theme: simplex
pdf_document: default
urlcolor: BrickRed
---
***
## Exercise 1
Consider a random variable $X$ that has a $t$ distribution with $7$ degrees of freedom. Calculate $P[X > 1.3]$.
```{r}
# your code here
```
***
## Exercise 2
Consider a random variable $Y$ that has a $t$ distribution with $9$ degrees of freedom. Find $c$ such that $P[X > c] = 0.025$.
```{r}
# your code here
```
***
## Exercise 3
For this Exercise, use the built-in `trees` dataset in `R`. Fit a simple linear regression model with `Girth` as the response and `Height` as the predictor. What is the p-value for testing $H_0: \beta_1 = 0$ vs $H_1: \beta_1 \neq 0$?
```{r}
# your code here
```
***
## Exercise 4
Continue using the SLR model you fit in Exercise 3. What is the length of a 90% confidence interval for $\beta_1$?
```{r}
# your code here
```
***
## Exercise 5
Continue using the SLR model you fit in Exercise 3. Calculate a 95% confidence interval for the mean tree girth of a tree that is 79 feet tall. Report the upper bound of this interval.
```{r}
# your code here
```
***
## Exercise 6
Consider a random variable $X$ that has a $t$ distribution with $5$ degrees of freedom. Calculate $P[|X| > 2.1]$.
```{r}
# your code here
```
***
## Exercise 7
Calculate the critical value used for a 90% confidence interval about the slope parameter of a simple linear regression model that is fit to 10 observations. (Your answer should be a positive value.)
```{r}
# your code here
```
***
## Exercise 8
Consider the true simple linear regression model
$$
Y_i = 5 + 4 x_i + \epsilon_i \qquad \epsilon_i \sim N(0, \sigma^2 = 4) \qquad i = 1, 2, \ldots 20
$$
Given $S_{xx} = 1.5$, calculate the probability of observing data according to this model, fitting the SLR model, and obtaining an estimate of the slope parameter greater than 4.2. In other words, calculate
$$
P[\hat{\beta}_1 > 4.2]
$$
```{r}
# your code here
```
***
## Exercise 9
For Exercises 9 - 13, use the `faithful` dataset, which is built into `R`.
Suppose we would like to predict the duration of an eruption of [the Old Faithful geyser](http://www.yellowstonepark.com/about-old-faithful/) in [Yellowstone National Park](https://en.wikipedia.org/wiki/Yellowstone_National_Park) based on the waiting time before an eruption. Fit a simple linear model in `R` that accomplishes this task.
What is the value of $\text{SE}[\hat{\beta}_1]$?
```{r}
# your code here
```
***
## Exercise 10
What is the value of the test statistic for testing $H_0: \beta_0 = 0$ vs $H_1: \beta_0 \neq 0$?
```{r}
# your code here
```
***
## Exercise 11
What is the value of the test statistic for testing $H_0: \beta_1 = 0$ vs $H_1: \beta_1 \neq 0$?
```{r}
# your code here
```
***
## Exercise 12
Test $H_0: \beta_1 = 0$ vs $H_1: \beta_1 \neq 0$ with $\alpha = 0.01$. What decision do you make?
- Fail to reject $H_0$
- Reject $H_0$
- Reject $H_1$
- Not enough information
```{r}
# your code here
```
***
## Exercise 13
Calculate a 90% confidence interval for $\beta_0$. Report the upper bound of this interval.
```{r}
# your code here
```
***
## Exercise 14
For this Exercise, use the `Orange` dataset, which is built into `R`.
Use a simple linear regression model to create a 90% confidence interval for the change in mean circumference of orange trees in millimeters when age is increased by 1 day. Report the lower bound of this interval.
```{r}
# your code here
```
***
## Exercise 15
For this Exercise, use the `Orange` dataset, which is built into `R`.
Use a simple linear regression model to create a 90% confidence interval for the mean circumference of orange trees in millimeters when the age is 250 days. Report the lower bound of this interval.
```{r}
# your code here
```
***
## Exercise 16
For this Exercise, use the `cats` dataset from the `MASS` package.
Use a simple linear regression model to create a 99% prediction interval for a cat's heart weight in grams if their body weight is 2.5 kilograms. Report the upper bound of this interval.
```{r}
library(MASS)
# your code here
```
***
## Exercise 17
Consider a 90% confidence interval for the mean response and a 90% prediction interval, both at the same $x$ value. Which interval is narrower?
- Confidence interval
- Prediction interval
- No enough information, it depends on the value of $x$
***
## Exercise 18
Suppose you obtain a 99% confidence interval for $\beta_1$ that is $(-0.4, 5.2)$. Now test $H_0: \beta_1 = 0$ vs $H_1: \beta_1 \neq 0$ with $\alpha = 0.01$. What decision do you make?
- Fail to reject $H_0$
- Reject $H_0$
- Reject $H_1$
- Not enough information
***
## Exercise 19
Suppose you test $H_0: \beta_1 = 0$ vs $H_1: \beta_1 \neq 0$ with $\alpha = 0.01$ and fail to reject $H_0$. Indicate all of the following that must always be true:
- There is no relationship between the response and the predictor.
- The probability of observing the estimated value of $\beta_1$ (or something more extreme) is greater than $0.01$ if we assume that $\beta_1 = 0$.
- The value of $\hat{\beta}_1$ is very small. For example, it could not be 1.2.
- The probability that $\beta_1 = 0$ is very high.
- We would also fail to reject at $\alpha = 0.05$.
***
## Exercise 20
Consider a 95% confidence interval for the mean response calculated at $x = 6$. If instead we calculate the interval at $x = 7$, mark each value that would change:
- Point Estimate
- Critical Value
- Standard Error
***