Hypothesis Testing

Question 1

What is type 1 error? What is it equal to? Give an example.

Answer 1

Type 1 error = reject H0 when it’s true
P(Type 1 error) = alpha - significance level
E.g. Convicting an innocent person

Question 2

What is a type 2 error? What is it equal to? Give an example

Answer 2

Type 2 error = accepting H0 when it’s false
P(Type 2 error) = 1 - Power

E.g. letting a guilt person go free

Question 3

What does it mean if we have a smaller significant level?

Answer 3

Smaller alpha = smaller type 1 error

We want to minimise the chance of convicting an innocent person = more likely to accept H0

Question 4

Normal CV for 10% in one tail

Answer 4

1.280

Question 5

Normal CV for 5% in one tail

Answer 5

1.645

Question 6

Normal CV for 2.5% in one tail

Answer 6

1.960

Question 7

Normal CV for 1% in one tail

Answer 7

2.320

Question 8

Normal CV for 0.5% in one tail

Answer 8

2.575

Question 9

If H0: M=70
H1: M<70
And alpha =5%, what is the CV if the underlying distribution is normal?

Answer 9

CV for 5% = 1.645

Since One sided alternative is less than:
CV = negative 1.645

Question 10

For basic hypothesis test: what test do we use if X is NOT normal but n>25/30?

Answer 10

N>23/30 = can invoke CLT

X bar is approximately normal with a mean M and variance sigma^2/n or S^2/n. Whether sigma^2 is known or not, we still do Z test.

Question 11

For basic hypothesis test: is X is normal but sigma^2 unknown, what test do we do?

Answer 11

T test using S^2 (sample variance)

T= X bar - M0 / sqrt S^2/n

Question 12

If the underlying series is a Bernoulli trial, and n>25/30, how is X bar distributed?

Answer 12

M = p and sigma^2 = p(1 - p)

X bar is approx normal with mean of p and p(1 - p) / n

Question 13

Diff in means: expectation and variance of X1 bar - X2 bar

Answer 13

E(X1 bar - X2 bar) = 0

V(X1 bar - X2 bar) = (sigma1) ^2 / n1 + (sigma2) ^2 / n2

As independent random samples covariance = 0

Question 14

Diff in means: if X1 & X2 Normal but population variances unknown

Answer 14

1. Test equality of variances
   If can assume sigma1 squared = sigma2 squared, pool variances & dof = n1 + n2 - 2

If sigmas not equal, do not pool and use complicated dof formula.

Question 15

What test do we do for equality of variance?

Answer 15

F test = S1 ^2 / S2 ^2 for S1 ^2 > S2 ^2

Question 16

What test do we do to test the variance of a distribution? What condition must hold for us to do this?

Answer 16

X MUST be normally distributed.

Chi-Squared test: Xn-1 = (n - 1) S^2 / sigma0 ^2

Question 17

What is matched pairs?

Answer 17

X1 & X2 are not independent - they are from the same sample.

Question 18

Variance of matched pairs vs random sample

Answer 18

Matched pairs: V(X1 - X2 all bar) = sigma1 ^2 + sigma2 ^2 - 2COV / n

Random sample: V(X1 bar - X2 bar) sigma1 ^2 / n1 + sigma2 ^2 / n2
As COV is positive, variance of matched pairs smaller so more efficient

Question 19

Matched pairs test statistic

Answer 19

Normal but sigma unknown –> t test

T = d bar - 0 / (sqrt Sd ^2 / n)

Question 20

What value of sigma do we assume if it is not given and we don’t have the sample variance either

Answer 20

0.5(0.5)

This gives the greatest value

Question 21

What is power?

Answer 21

Power = 1 - Type 2 error

Power = P( reject H0 given H0 false)
Probability of correctly rejected the null.

Question 22

3 stages to working out power

Answer 22

1. What is our CV - what distribution? 2-sided = +- CV
2. Convert CV to our null distribution = Xc
3. Under MT, work out probability Z is > Xc if H1 is >

Question 23

Can power be in both tails? When?

Answer 23

If we have a 2 sided alternative, we use +- CV which gives us two Xc s for power.

Question 24

What values can power lie between?

Answer 24

Alpha = significant level and 100%

Question 25

When is power = significance level?

Answer 25

When the null distribution and true distributions are the same.

Question 26

What is ANOVA? Why use it?

Answer 26

ANOVA = analysis of variance

Allows us to test equality of means among > 2 groups instead of doing separate tests which makes it inevitable that we reject H0 at some point

Question 27

3 assumptions for ANOVA

Answer 27

1. X normally distributed
2. Variances of each group the same
3. Independent random samples

Question 28

Test statistic for ANOVA

Answer 28

F = [BSS / (k - 1)] / [WSS / (n - k)]

Question 29

For ANOVA how do we calculate the overall sample mean X bar?

Answer 29

X bar = X1bar n1 + X2bar n2 + … + Xkbar nk / n

We weight the sample Mean for each group by the number in the group.

Question 30

DOF for BSS for ANOVA

Answer 30

K - 1 where k is the number of groups

Question 31

DOF for WSS for ANOVA

Answer 31

n - k where n is overall sample size and k is the number of groups

Question 32

Diff in means where we are looking at proportions, how do we work out variances?

Answer 32

S1^2 & S2^2 worked out using X1bar and X2 bar proportions success x proportion failure.

Since P1=P2 under H0, pool sample variances so find S0^2 & use this.

Question 33

If n>25/30 and are using sample PROPORTIONS, what distribution do we use for difference in means and if sigmas unknown, what variance?

Answer 33

Invoke CLT = approx normal.

Use a POOLED sample variance: (n1 - 1)proportion 1 + (n2 - 1)proportion 2 / n1 + n2 - 2