Confidence Intervals

Question 1

What do CI work out?

Answer 1

A range of likely values for the true but unknown population parameter based on sample data, together with a measure of confidence / likelihood that the range contains the true value.

Question 2

A CI is wider if…

Answer 2

1) more confidence
2) greater variance
3) smaller sample size n

Question 3

CV for 90% CI for normal

Answer 3

+- 1.645

Question 4

CV for 95% CI for normal

Answer 4

+- 1.960

Question 5

CV for 99% CI for normal

Answer 5

+- 2.575

Question 6

CI for M is given by…

Answer 6

M belongs to [X bar +-CV Se(X bar)]

Where Standard error = sqrt sigma^2 or S^2 / n

Question 7

What does a 90% CI for M mean?

Answer 7

90% of occasions M is inside the interval. 10% of the time we find M < or > the limits.

Question 8

Finding CI for M: if X is normal but sigma^2 unknown, from what distribution do we get the CV from?

Answer 8

T distribution with n-1 DOF.

Question 9

If the underlying distribution is a Bernoulli trial, what distribution do we get the CV from for CI for M?

Answer 9

Usually would use M=p and sigma^2 = p(1-p) but obviously we don’t know p otherwise we wouldn’t be constructing a CI for M as its exact value would be known. So we use S^2 (p(1-p) based on sample data) and as long as n>25/30 invoke CLT = approx normal.

Question 10

If n=21 and we do not know the underlying distribution, what can we do?

Answer 10

1) do nothing

2) assume normality - then if sigma^2 unknown = t distribution

Question 11

What do we assume before constructing difference in means CI?

Answer 11

That X1 and X2 are independent random samples (so covariance = 0)

Question 12

CI for M1 - M2

Answer 12

(X1 bar - X2 bar) +- CV Se(X1 bar - X2 bar)

Standard error = sqrt (sigma1^2 / n1 + sigma2^2 / n2) if sigmas known.

Question 13

Suppose distribution of X1 & X2 is unknown, sigmas unknown, n1 & n2 > 25/30. What distribution?

Answer 13

Invoke CLT = approx normal.

CV from Z tables.

Use S1^2 and S2^2.

Question 14

Suppose X1 & X2 normal, sigmas unknown. What test?

Answer 14

T test.
If sigmas equal, pool sample variance & CV from t with DOF = n1+n2 - 2

If sigmas not equal, do not pool variances, CV from t with DOF see complex formula.

Question 15

What condition must be satisfied to construct CI for the variance of a distribution?

Answer 15

X MUST be normally distributed

Question 16

If the underlying distribution of a random variable is normal, what is the distribution of its sample variance?

Answer 16

Chi squared with DOF = n - 1

Question 17

How do we split the CI proportions for variance CI?

Answer 17

Symmetrically - split equally. As otherwise it’d be inefficient since chi squared share varies as DOF change so we’d have to work out the optimal proportions for every Q. The CI with equal proportions is easier but doesn’t minimise the interval range.

Question 18

CI for population variance

Answer 18

[(N-1) S^2 / CV-, (N - 1) S^2 / CV+)

Question 19

How do we find the CVs for 95% CI for population variance?

Answer 19

Chi squared n
Alpha / 2 = 0.025
Look up CV for 0.025 and n
Look up CV for (1-0.025)=0.975 and n

Question 20

CI for matched pairs M1 - M2

Answer 20

M1 - M2 = [d Bar +- CV (sqrt Sd^2/n)]

CV from t distribution if X1&X2 normal but we don’t know sigma d.

Question 21

If the underlying distribution is a Bernoulli trial, we work out a CI for…

Answer 21

The population PROPORTION I.e. Divide by n