sampling distribution and estimation

Question 1

describe the sample mean ?

Answer 1

it is an estimate

a random variable drawn from the population

Question 2

name the different types of distribution ?

Answer 2

• normal
• binomial
• chi squared
• f distribution
• t distribution

Question 3

describe a normally distributed sample

Answer 3

• sample size of n
• sample mean xbar = sum of x / n
• variance = sigma(sd)^2 / n
• population mean is mew
• z transformation = sample mean - popmean / st,e
• standard error = root ( sigma^2 /n)

Question 4

what is central limit theorem ?

Answer 4

the theory that as long as n > 25, you can use the methods you would use on normally distributed data to analyse it.

Question 5

what is ?

a. an unbiased estimator ?
b. a biased estimator ?

Answer 5

a. an estimator ( for the mean ) that = the true unknown parameter
b. doesnt equal the true value

(mean strays from the centre of the distribution)

Question 6

for a normally distributed sample, how do you estimate the population variance ?

Answer 6

the sample variance is an unbiased estimator so we calculate this:

s^2 = sum of ( xi - sampmean)^2 / n - 1 = sigma^2

Question 7

what is high versus low precision ?

Answer 7

high precision = low variance
low precision = high variance

the steeper the distribution the lower the variance and the more preferred

Question 8

do we prefer data to be precise or unbiased ?

Answer 8

generally we prefer unbiased data

Question 9

for a sample with either N>25 or norm dist, how do you construct a confidence interval ?

Answer 9

first find the critical values for the size interval you need

transform these into critical values of the sampling mean

c L = sample mean - critval . sigma/root n
c U = “ but + not -

Question 10

for a sample with either N>25 or norm dis, how do you estimate a population proportion ?

Answer 10

• sample size n
• sample prop p
• r no of successes
• pop prop = pie

we us binomial because there are only two options for outcomes, there fore the sample size must be > 25

p = r/n 
p = mean
variance = pie (1-pie)  / n

cU/L = p +/- Zcv.standardd (which is root( (p.(1-p)) / n)

Question 11

what is the t distribution ?

Answer 11

it is very similar to normal distribution but it has one parameter V - degrees of freedom

Question 12

how do you calculate confidence intervals or anything with t distribution ?

Answer 12

you calculated the same and normal but with different cvs.

v = n-1

t = sample mean - population mean / root (sample variance /n )

Question 13

how do you conduct a hypothesis test ?

Answer 13

1) formulate the null and alternative hypotheses
2) chose the level of significance of the test
3) look up the correct critical values and set rejection region
4) calculate the test statistic
5) compares test stat to re region and make decision about hypotheses acceptance

Question 14

how do you calculate the test statistic for

a. std normal
b. t dist

Answer 14

a. z = x - population mean / standard deviation

b. t = sample mean - population mean / root (sample variance /n )

Question 15

what is type || error probability ?

Answer 15

the probability that we fail to reject the null hypothesis even when its true

the prob is equal to the area under the H1 is true distribution, to the left of the upper cv of the H0 is true distribution

Question 16

what is the power if a test ?

Answer 16

the probability of rejecting the null when its false = 1 - B

Question 17

the more powerful the test the better, so how do we increase the power ?

Answer 17

• increase the sample size ( less overlap - area is smaller)
• avoid situations where effect sizes is small ( effect size is the size of difference between actual and hypothesised means )
• smaller sample variance ( when possible but we normally don’t get the choice in econ )

Question 18

what does statistically significant mean and does it mean economically important

Answer 18

it means that we reject the null hypothesis

no.

the difference can be small but if the sample size is large enough the difference would still be significant and yet it would not be statistically important.

Question 19

how do you test a population proportion ?

Answer 19

always a Z test, we rely on the binomial approximation for a normal distribution.

therefore:MUST n.p > 5 , n. (1-P) > 5

find the significance level

critical values and set the rejection region. cv = std normal ones from Z table

test statistic Z* = P - pie 0 / (root ( pie (1 - pie)/n ))

Question 20

how do you test the difference of two population means ?

Answer 20

same principle applies as before.
we only do the calculations when we know what the population variances are.

find the hypothesis which is always that the difference is not significant.

find the significance level

find the cv ( on the Z table ) and set the rejection region

calculate the test statistic and compare

test stat Z = ( sample mean1 -sm2 ) . (popmean1 - pm2) /
root( pv/n + pv/n)

Question 21

what is the chi squared distribution ?

Answer 21

it is the distribution of the sum of squares of independent random variables.

Question 22

where does the X^2 distribution lie ?

Answer 22

only in the posative domain

Question 23

how many parameters does the X^2 distribution have ?

Answer 23

one, K degrees of freedom.

Question 24

how do you calculate the…… for the X^2 distribution ?

a. sample variance
b. expected value ( mean )
c. variance

Answer 24

a. s^2 = the sum of ( (Xi - samplemean)^2 ) / n-1
b. k
c. 2k

Question 25

as n reaches…….what….. does the x^2 distribution move to normal by central limit theory ?

Answer 25

infinity

Question 26

how do we calculate a confidence interval for X^2 distribution ?

Answer 26

we can’t use a regular Ci because the interval is not symmetric as varience can’t be lower than 0

we transform S^2 so it has X^2 dist
(n-1) .s^2 / sigma^2

then we find the critical values
upper cv - one of the tails % in the t table
lower cv - 1 - one of the tails % in the t table

actual cvalues

lower (n-1) .s^2 / lowercv

upper (n-1) .s^2 / uppercv

this whole experiment only holds if the population is normally distributed

BUT IF THE SAMPLE SIZE IS > 100 X2 IS A VERY GOOD ESTIMATE SO WE USE NORMAL CV AND TRANSFORM THEM

0.5(zcv +/- root(2v -1 )^2

Question 27

how many sides do X^2 hypothesis tests have ?

Answer 27

always only look at one, whether or not its two tailed or not.

Question 28

how do you calculate a test statistic for a hypothesis test in a X^2 distribution ?

Answer 28

X^2 = sum of ( ( O - E )^2 ) / E

all E for categories must be %, if they aren’t you must combine categories until they are.

Question 29

how do you conduct a test for independence ?

Answer 29

set up hypotheses
chose significance level 
find the critical values 
calculate the test statistic =  sum of ( ( O - E )^2 ) / E
make a decision 

need V for finding cvs

V = (r-1).(c-1)

Question 30

what is the F distribution ?

Answer 30

the ration of two independent X^2 distributed random variables, both scaled by their degrees of freedom.

F = (x1/v1) / (x2/v2)

under the null hypothesis th population var is the same and as the two variables are normally distributed

f = S1^2 / s2^2

Question 31

how do you conduct a variance ratio test ?

Answer 31

set up hypotheses - always that the pop vars are =
set up significance level
find critical value in F table we are only interested in one
calculate the test stat - f = S1^2 / s2^2
decision

Question 32

how do you calculate the expected value of a section in the table for a chi squared test ?

Answer 32

( row total x column total ) / grand total