7. Hypothesis Testing

Question 1

Hypothesis test: z test

Answer 1

Z(obs) = x̅-μ0 / σ/sqrt(n)

p-value = 2(1 - Φ|Zobs| )

Question 2

Hypothesis test: t test

Answer 2

Use this when variance is unknown.
t(obs) = x̅-μ0 / s/sqrt(n)
p-value = 2(1 - Φ|Zobs| )

Question 3

R code for hypothesis test using standard normal distribution.

Answer 3

pnorm( z(obs), 0, 1)

Question 4

R code for hypothesis test using t distribution.

Answer 4

pt( t(obs), n-1)

Question 5

Type 1 error

Answer 5

Reject H0 when it is true.
p(type 1 error) = α
where α is the significance level being tested.

Question 6

Type 2 error

Answer 6

Accept H0 when it is false.

p(type 2 test) = 1 - power

Question 7

What is the power of a hypothesis test?

Answer 7

power = 1 - p(type 2 error)

= Φ(-Zcrit - (μ-μ0)/ σ/sqrt(n) ) +1 - Φ(Zcrit - (μ-μ0)/ σ/sqrt(n) )

Question 8

Testing with a fixed significance level, α. When do we reject H0?

Answer 8

We reject H0 if

|Z(obs)| > Φ^-1( 1 - α/2)

Question 9

Two sample hypothesis tests when the samples are dependent.

Answer 9

Define a new random variable: Di = Xi - Yi. Then Di and Dj are independent.
(d-bar) = (x-bar) - (y-bar).
Then continue with t as normal.

Question 10

Two sample hypothesis tests when the samples are independent and variance is unknown.

Answer 10

We can work with the raw data. We compare:
T = (Xbar - Ybar) - μ0 / sqrt( Sx^2/nx + Sy^2/ny)
with the T distribution it would have if H0 were true.

Question 11

Two sample confidence interval for μ when the samples are independent and variance is unknown.

Answer 11

(xbar - ybar) +- T(v, 1-α/2) sqrt(S x^2/nx + Sy^2/ny)

Question 12

Plus four / Wilson confidence interval for a proportion.

Answer 12

Replace p^=x/n with p~=(x+2) / (n+4)