Confidence Intervals Flashcards
What do CI work out?
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.
A CI is wider if…
1) more confidence
2) greater variance
3) smaller sample size n
CV for 90% CI for normal
+- 1.645
CV for 95% CI for normal
+- 1.960
CV for 99% CI for normal
+- 2.575
CI for M is given by…
M belongs to [X bar +-CV Se(X bar)]
Where Standard error = sqrt sigma^2 or S^2 / n
What does a 90% CI for M mean?
90% of occasions M is inside the interval. 10% of the time we find M < or > the limits.
Finding CI for M: if X is normal but sigma^2 unknown, from what distribution do we get the CV from?
T distribution with n-1 DOF.
If the underlying distribution is a Bernoulli trial, what distribution do we get the CV from for CI for M?
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.
If n=21 and we do not know the underlying distribution, what can we do?
1) do nothing
2) assume normality - then if sigma^2 unknown = t distribution
What do we assume before constructing difference in means CI?
That X1 and X2 are independent random samples (so covariance = 0)
CI for M1 - M2
(X1 bar - X2 bar) +- CV Se(X1 bar - X2 bar)
Standard error = sqrt (sigma1^2 / n1 + sigma2^2 / n2) if sigmas known.
Suppose distribution of X1 & X2 is unknown, sigmas unknown, n1 & n2 > 25/30. What distribution?
Invoke CLT = approx normal.
CV from Z tables.
Use S1^2 and S2^2.
Suppose X1 & X2 normal, sigmas unknown. What test?
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.
What condition must be satisfied to construct CI for the variance of a distribution?
X MUST be normally distributed
If the underlying distribution of a random variable is normal, what is the distribution of its sample variance?
Chi squared with DOF = n - 1
How do we split the CI proportions for variance CI?
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.
CI for population variance
[(N-1) S^2 / CV-, (N - 1) S^2 / CV+)
How do we find the CVs for 95% CI for population variance?
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
CI for matched pairs M1 - M2
M1 - M2 = [d Bar +- CV (sqrt Sd^2/n)]
CV from t distribution if X1&X2 normal but we don’t know sigma d.
If the underlying distribution is a Bernoulli trial, we work out a CI for…
The population PROPORTION I.e. Divide by n
