Math Answers

Show that | z + w |^2 − | z − w |^2 = 4Re (zŵ).

If z=cosθ+i sinθ,prove that

zn + z−n = 2cos(nθ), z^n − z^−n = 2i sin(nθ).

Apply De Moivre’s formula to express cos 4θ and sin 4θ in terms of cos θ and sin θ.

Determine the distance between 2 − i and 3 + i on the complex plane.

How many pairs of dance partners can be selected from a group of 12 women

and 20 men?

Find the area enclosed by the curve r = 5cosθ

Timothy Wilson, a psychology professor at the University of Virginia, studies the effect of "personal narratives" on people's behavior. For example, if the "story" you have about your academic ability is positive, you do better in school than if your "story" is negative. Something as simple as hearing older students describe overcoming challenges similar to yours can help you change your story and improve your performance.

A) Suppose you have 40 college freshmen who have volunteered to be part of an experiment. Design a completely randomized experiment that tests the hypothesis that hearing older students talk about overcoming challenges improves academic performance. Be sure your design addresses how randomization will be incorporated.

B) Can this experimental design be carried out in a double blind manner? Explain.

The sales of Lexus automobiles in the Detroit area follow a Poisson distribution with a mean of 5 per day. (i) What is the probability that no Lexus is sold on a particular day? (ii) What is the probability that at least two Lexus is sold on consecutive 3 days? 

n All maximum possible random samples of size 4 are drawn from the finite population comprising of X=2,4 a) Verify central limit theorem (CLT) in this random experiment b) Between what two sample means would you expect the middle 75% of sample means to fall c) Estimate population mean at 90% C

4. Let A and B be sets. Prove the commutative laws from Table 1 by showing that

a) A ∪ B = B ∪ A.

b) A ∩ B = B ∩ A.