---
title: STAT 3340 Assignment 2, fall 2026 - due Sunday, October 4 , 11:59 PM
author: "Your name here"
date: 'Banner:  B00??????'
output: 
  pdf_document: default
  word_document: default
---


1. The length of a species of fish is to be represented as a function
of the fish's age and the water temperature.  The fish are kept in tanks
at 25, 27, 29 and 31 degrees Celsius.  The following reads some data
on the age, water temperature, and length of fish, and fits a number of
regression models.


```{r}
data=read.csv("http://www.mathstat.dal.ca/~bsmith/stat3340/Data/fish.csv",header=T)
age=data[,1]
temp=data[,2]
length=data[,3]
lm0=lm(length~1)
lm1=lm(length~age+temp)
lm2=lm(length~age+temp+age:temp)
lm3=lm(length~age+temp+I(age^2)+I(temp^2))
lm4=lm(length~age+temp+age:temp+I(age^2)+I(temp^2))
```

   + a) compare models lm0 and lm2.

   + a i)  Write down the linear regression models associated with lm0 and lm2.
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   + a ii) What are the associated null and alternative hypotheses when comparing the two models.
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   + a iii) Use the anova command to compare the outputs lm0 and lm2.
```{r}
#enter your R commands here.
```


   + a iv)  What are the observed value of $F$ and the p-value.

\newpage
   + b) Compare two quadratic models, one which includes an interaction term, and
        the other which doesn't.

   + b i)  Write down the $full$ and $reduced$ regression models.
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   + b ii) What are the associated null and alternative hypotheses.
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   + b iii) Use the anova command to compare the outputs for the full and reduced models.
   
```{r}
#enter your R commands here.
```

   + b iv)  What are the observed value of $F$ and the p-value.

\newpage
2.  An experiment was carried out to assess the yield of 4 different crop
types on yield.  The data are entered into R as follows:

```{r}
yield=c(123,128,166,151,156,150,178,125,112,174,187,117,100,116,153,155,
168,109,195,158,135,175,140,167,130,132,145,183,176,120,159,142,120,187,
131,167,155,184,126,168,156,186,185,175,180,138,206,173,147,178,188,154,
146,176,165,191,193,190,188,169)

crop=as.factor(rep(c("W","C","S","R"),15))
```

A boxplot of yield vs crop type is as follows:

```{r}
boxplot(yield ~ crop)
IW=ifelse(crop=="W", 1,0)
```

2a) Write down a multiple regression model corresponding to a one way
analysis of variance of yield as a function of crop type.

(Hint:   Define indicator variables for the different crop types, as done above for crop type W.  Your  multiple   regression model should include three of the indicator variables as predictors.) 


\newpage
2b) Fit the regression model in R using the "lm" command, and show the summary output.

```{r}
#lm.out=lm( ...
#summary(lm.out

```

2c)  What is the observed value of the F statistic?
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(You can check your results using "anova(lm(yield$\sim$crop))".   


  

\newpage
3. 
An experiment was carried out to assess the effect of diet on weight loss.

Five mice were put on each of three diets. At the beginning of the
experiment, each animal's weight was measured, and recorded recorded
as the variable $x$. After 3 months on diet, the animal's weight  was
measured again, and recorded as $y$.

Write down a single multiple regression model which allows for
different slopes and different intercepts between y and x for each
of the three diets. That is, the one multiple regression model
should allow for 3 different linear regressions of y on x, one
regression for each diet, and allowing for the 3 regression lines
to have different slopes and intercepts.
  
3a) Carefully define each variable to be used in the regression model.

(Hint:  you'll need to define appropriate indicator variables to code for
the different diets.)
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3b) In terms of your model parameters, state the null and alternative
hypotheses to be used when testing that the slopes of the 3 regression
lines are the same, but allowing for the intercepts to be different.



\newpage

4. An experiment was carried out to assess the effect of sex (Male and Female),
and diet type (I, II or III) on weight loss.  Five mice were randomly assigned
to each each combination of sex and diet.  The outcome
variable $y$ was the individual's change in weight after 3 months on the diet.

4a) Write down a single linear regression model that can be used to fit a two way analysis
of variance model for weight change, which allows for an interaction between
sex and diet type. 

Carefully define each variable to be used in the regression.

(Hint:  you'll need to define appropriate indicator variables to code for
sex and diet.)
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4b) In terms of your model parameters, state the null and alternative hypotheses used
when testing for the presence of an interaction.
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\newpage
    
5. Four regression models were fit to a dataset containing 102 observations.
The model and error sum of squares for each of the models are included in the following table.

\begin{center}
\begin{tabular}{c | c}
    Model & SSE \\ \hline
    \\
    $y=\beta_0 + \epsilon$ & 208\\
    \\
    $y=\beta_0 + \beta_1 x_1 + \epsilon$ & 200\\
    \\
    $y=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon$ & 198\\
    \\
    $y=\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2 + \epsilon$ & 196 \\
\end{tabular}
\end{center}
      


5a) When testing the hypothesis $H_0: \beta_3 = 0$, what is the observed value of $F$ and what are the numerator and denominator degrees of freedom? 

Hint: Use the partial F test where $F=\frac{(SSE_{red}-SSE_{full})/r}{SSE_{full}/(n-p)}$.

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5b)  When testing the hypothesis $H_0: \beta_1 = \beta_2 = \beta_3 = 0$, 
    what is the observed value of $F$ and what are the numerator and denominator degrees of freedom?