Inferential Stats

Question 1

For a single sample,
What are the criteria to use z?

Answer 1

Large sample size, n ≥ 30
Population variance can be known or unknown.

OR

Small sample size, n < 30
Population variance known

Question 2

For a single sample,
What are the criteria to use t?

Answer 2

Small sample size, n < 30
AND
Population variance is unknown.

Question 3

Assumption for:
Two independent populations mean test using t

Answer 3

1. Underlaying data are normally distributed
2. Population variance of both populations are the same

Question 4

Assumptions for:
Paired samples mean test using t

Answer 4

1. Population differences are normally distributed
2. Population variance of both populations are the same

Question 5

When to use
x̅ – y̅
VS
d̅

Answer 5

x̅ – y̅ : two independent samples

d̅ : two paired samples

Question 6

A question of two paired samples wants you to find confidence intervals of μ_x–μ_y, will you use find x̅ – y̅ or d̅

Answer 6

d̅ as it is paired sample.

Question 7

It is believed that the means of two population differs.
Therefore,
H₀:D̅ = 0
H₁:D̅ ≠ 0

True or false?

Answer 7

False, hypothesis must refer to population; D̅ is mean of samples differences. μ_x–μ_y OR μ_d are acceptable.

Question 8

5 samples are taken before and after an adjustment.
A statistician uses paired samples t-test using two distinct sample variances. Comment on his method.

Answer 8

He should be fired.
Populations tested using t-test are always assumed to share a common variance.

Question 9

When doing mean tests using z, or finding confidence interval of mean using z, is it necessary for a sample to be normally distributed?

Answer 9

No. If the sample size of the population is sufficiently large (n > 30), due to central limit theorem, the mean of sample, X bar, appears to be/is approximately normally distributed.

Question 10

A random sample of 50 batteries, claimed to be identical, taken from factories A and B each, are tested for the mean lives. Should you find the common variance and do paired test?

Answer 10

No and no.
It is not known whether they share a common variance and the batteries are not paired in any way meaningful

Question 11

It is claimed that A and B are the same.
Confidence interval is given to be (-15,8).

Does the confidence interval support the claim?

Answer 11

Yes.
If A and B have no difference,
μ_a–μ_b
or
μ_d
was claimed to = 0,

And 0 is within the interval.