STAT4710J Homework 1 Solution

1. Fundamental Linear Algebra
Ben, Tom, and Amy are shopping for fruit at a grocery store. A fruit bowl contains some fruit and the price of fruit bowl is the total price of all of its individual fruit.
The store has apples for $2, bananas for $1, and oranges for $4. The price of each of these can be written in a vector:

The store sells the following fruit bowls:
1. 2 of each fruit
2. 5 apples and 8 bananas
3. 2 bananas and 3 oranges
4. 10 oranges
(a) Define a matrix B such that
B→−v
evaluates to a length 4 column vector containing the price of each fruit bowl. The first entry of the result should be the cost of fruit bowl 1, the second entry the cost of fruit bowl 2, etc.
(b) Ben, Tom, and Amy make the following purchases:
• Ben buys 2 fruit bowl 1s and 1 fruit bowl 2.
• Tom buys 1 of each fruit bowl.
• Amy buys 10 fruit bowl 4s
Define a matrix A such that the matrix expression
AB→−v
evaluates to a length 3 column vector containing how much each of them spent. The first entry of the result should be the total amount spent by Ben, the second entry the amount sent by Tom, etc.
(c) Let’s suppose the store changes their fruit prices, but you don’t know what they changed their prices to. Ben, Tom, and Amy buy the same quantity of fruit baskets and the number of fruit in each basket is the same, but now they each spent these amounts:

In terms of A, B, and →−x , determine (the new prices of each fruit).
2. Calculus
Let .
(a) Show that σ(−x) = 1 − σ(x)
(b) show that the derivative can be written as:
dxd σ(x) = σ(x)(1 − σ(x))
(c) Make a plot for σ(x) and ) in the same coordinate system for x ∈ [−5,5]
3. Minimization
Consider the function . In this scenario, suppose that our data points x1,x2,...,xn are fixed, and that c is the only variable.
Using calculus, determine the value of c that minimizes f(c). You must justify that this is indeed a minimum, and not a maximum.
4. Probability
Only 1% of 40-year-old women who participate in a routine mammography test have breast cancer. 80% of women who have breast cancer will test positive, but 9.6% of women who don’t have breast cancer will also get positive tests.
Suppose we know that a woman of this age tested positive in a routine screening. What is the probability that she actually has breast cancer? (Note: You must show all of your work, and also simplify your final answer to 3 decimal places.)
5. Statistics
For inferential purposes, we choose to bootstrap this sample 500,000 times. That is, we simulate the act of re-sampling (with replacement) 200 students from our observed sample, and each time we record the number of Canadians in our re-sample. We provide a histogram of the sampling distribution below.

What is the standard deviation of the sampling distribution shown above? Select the closest option below, and explain your answer.
a 1.5 b 6.1 c 12.4 d 10.1
Hint: While it is possible to calculate the answer, the histogram has all of the information you need.
6. Inference
a What is the population and sample in this problem
b What is your null and alternative hypothesis? (please express them with both verbal description and equations)
c If we are looking at 100 respondents from the Literary Digest’s poll (i.e. 43% out of 100 vote for Roosevelt). Which hypothesis will you be in favor of with a significance level of
1%? (Please show your work)