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MTH116 Statistics

Ql. Interpreting coefficients for categorical variables Suppose one is studying growth rates among young land snails; to do so, they gather rt recently-hatched snailg from two different species (A and B), determine their age (days), and measure the diameter of their shell aperture (mm) They consider the following model for the data: diameter, = $ + thage, + /32 1{species, = + /33 (age, x 1.{ species, = + The growth rate is defined as the change in mean diameter with age. 

i. Which species is the reference? What is the growth rate for species B?
iii. What is the growth rate for species A?
iv. Interpret each parameter i30, $i, $2, 03.
v. How would you test whether the growth rates differ between species? Write the null and alternative hypotheses in terms of the appropriate model parameters.
vi. Suppose you learned that snails were born without shells (they aren't — this is a purely hypothetical conjecture); how would you adjust the model, if at all? 

Q2. One categorical predictor (You may type or write on a paper) Consider a linear model with a single categorical predictor that distinguishes two groups: ,610 +$11{groups 2} + Here, the variable groups can be either 1 or 2. In effect, this is a model for a single variable (Y) measured for individuals in two groups. Suppose there are nt observations in group 1, and rts observations in group 2; let lesi denote the jth observation in group i. 
i. What is EY for group 1? For group 2?
ii. Assume aithout loss of generality that observations are indexed so that all observations in group 1 are followed by all observations in group 2. In other words: (Yrt \ In, I Yi 1%, Y21.

Write the model in matrix form. iii. Find x'x and (x'x) 1. iv. Find x'Y. Write the matrix product in terms of nt, n2, and the group means 31. and Y2. v. Find R. (Hint your answer should be very simple and make intuitive sense given (i).) vi. Find tang. vii. Show that the partial significance test of 31 is equivalent to the equal variance two-sample t-test.

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