Hypothesis Testing

Question 1

Hypothesis testing formula for standard normal (SLIGHTLY DIFF)

Answer 1

Z = Xbar - μ / (√σ²/n)

~N(0,1)

Question 2

What happens when we replace σ² with a sample estimator

Answer 2

Swap it for s² and becomes a T distribution not Z! Which mean degrees of freedom changes to T(n-1)

Question 3

Error 1 in hypothesis testing, and notation for the probability of it occuring

Answer 3

Rejecting a hypothesis when it is true.

Notated by ∞

Question 4

Error 2, and its notation.

Answer 4

Not rejecting a false hypothesis

Notation - β (probability of making error 2)

Question 5

So how to reduce type 1 error?

Answer 5

Keep level of significance ∞ low e.g 0.05 (5%) , effectively meaning we are prepared to accept a 5% chance of committing a TYPE 1 ERROR

Question 6

Technical appendix for properties of the expected value operator (5)

What are we assuming

Answer 6

Assuming a and b are constants

1. E(a) = a e.g E(Xbar) = Xbar
2. E(ax) = aE(X) i.e we can take constants out the bracket
3. E(aX+b) = E(aX) + E(b) = aE(X) + b (Upholds 1st property)
4. E(X+Y) = E(X) + e(Y)
5. 𝐸(𝑋𝑌) ≠ 𝐸(𝑋)𝐸(𝑌) unless 𝑋 and 𝑌 are independent

Question 7

Properties of variance

What assumption do we have to make

Answer 7

Assume a and b constant

1. Var(b) = 0 as b is a constant and its value does not change
2. Var(X+b