chapter 8 Estimation

Question 1

when is a esitimator for a paramater say theta unbiased

Answer 1

it is unbiased when E(theta hat) = theta. theta represents the real value of the parameter

Question 2

How to calculate bias of a point estimator estimating parameter theta

Answer 2

B(theta hat) = E[theta hat] - theta. thus how far of the value is of the actual value of theta

Question 3

what is the MSE mean squar error of a point esitmator and what does it mean

Answer 3

Describes the goodness of a estimator taking into account the biase and the variance of the esitmator closer to 0 is better . MSE(theta hat) = E((teta^ - teta)^2)

Question 4

what is the MSE of a estimator when variance and bias is known

Answer 4

MSE(thete^) = v(teta^) + B(theta^)^2

Question 5

what is the error of estimation

Answer 5

distance between the estimator and target parameter thus epsilon = |theta^ - theta|

Question 6

how to estimate a confidence interval for u when the sample size is small.

Answer 6

calculate the sample mean x bar and sample variance s^2. Then the look up values for t distrubution that matches the confidence interval specified. THe degrres of freedom is specified by n-1 where n is samle size. then solve the enequality to isolate the true mean and get the bounds where this value falls within

Question 7

how to estimate the difference between two sample means x bar and y bar if variance is unknown. population variances must be the same.

Answer 7

First calculate the mean and variance for both observations. Then calculate the joint variance Sp^2 = (n-1)*S1^2 + (n2-1)^S2^2 / n1+n2 -2 . basically a weighted average of the two variances. Then this is also t distributed but now with n1+n2 - 2 df. Remember this has the n as sqrt(1/n1+1/n2);

Question 8

estimate a confidence inteval between the differance between two sample means but now assuming that the sample variances are different.

Answer 8

now the variable becomes
X1bar - X2bar -(u1-u2) / sqrt( S1^2/n1 + S2^2/n2)
has a t distrubution with v degrees of freedom where v is estimated and rounded down using the welch-statterweight equasion.

Question 9

What is the welch-statterwait equasion

Answer 9

This is for estimating the degrees of freedom when difference between two means want to be estimated but the variance of the two smaples are differnent.
df = (S1^2/n1 + S2^2/n2)^2 / S1^4/n1^2(n1-1) + S2^2/n2^2(n2-1) and then we round down.

Question 10

How to construct a confidence interval for the population variance

Answer 10

First calculate the sample variance S^2 and smaple mean x bar the the random variable is
(n-1) S^2 / sigma^2 is X^2 kai squared distrubuted with n-1 df.

Question 11

How to construct a condidence interval for the ratio of two population variances.

Answer 11

come back later