Lec 4 notes

Question 1

What does randomization do

Answer 1

What does randomization do?
1. removes selection bias
2. establish causality

Question 2

sampling variability

Answer 2

Sampling variability

Population mean is 100
But from sample A,B,C; they are 101, 102, 99
The sample mean varies from sample to sample and does not come out exactly 100
Here, the x bar (mean) varies from sample to sample and that it did not come out to exactly 100 (the pop mean)

Question 3

Sampling distribution

Answer 3

Sampling distribution
if you sample over and over, it zeroes in on what you are looking for (ie pop mean)

Sample distribution of the mean = pop mean

µX bar = mean of the sampling distribution of the random variable X bar

Question 4

relationship b/w sample mean and population mean

unbiasedness

Answer 4

sample mean is a point estimate of the population mean

Unbiasedness: the mean of the sampling distribution is equal to the population parameter

Question 5

standard error of the mean
- type of staistic
- is it data
- sample size and SE relationship
- SE is a measure of ???

Answer 5

standard error of the mean (sem) or standard error: it is NOT a descriptive stat; it is a summary statistics of the sampling distribution

It is NOT data b/c if has only ONE point in the data

The larger the sample size, the more precise our estimate of µ will be

Also, larger sample size -> smaller SE

SE indicates the variability of the estimate

Question 6

bias
How can we reduce bias?

variability
How can we reduce variability?

Answer 6

Bias
unbiased if the mean of its sampling distribution is equal to the true value of the corresponding parameter in the pop; bias = opp

Randomization techniques reduce bias

Variability: Concerns the spread of the sampling distribution
o It is reduced by increasing the sample size, n

Question 7

Central Limit Theorem

How big does n need to be?
Exception

Answer 7

Central limit theorem

the distribution of sample means approximates a normal distribution N(µ, σ2/n) [w/ mean µ and variance σ2/n] if we increase the sample size (n -> infinity),

How big does n need to be?
It depends on the pop distribution, but a guideline is n > 10

BUT, if the pop comes from an extremely skewed distribution, n may need to be larger

Answer 8

How do we interpret a confidence interval?
- For 95% CI:
- In repeated random sample from the pop, 95% of the confidence intervals computed from the samples will contain the true value of the pop parameter (µ for example)

Answer 9

• Difficulty is we cannot PROVE the truth of any H
• If our data are NOT consistent w/ H0 being true, we reject it in favor of H1(although that doesn’t mean H1 is true either)
• If the data are consistent w/ H0, we cannot reject H0, which is not the same thing as accepting H0 (test stat is closer to 0)
• Example
• Some Hypotheses
  o H0: µ = µ0 vs H1: µ = µ1
  o H0: µ ≤ µ0 vs H1: µ > µ0 (one sample)
  o H0: µ = µ0 vs H1: µ ≠ µ0 (two sample)

Answer 10

• Definitions
  o TI error: occurs when H0 is rejected when in fact it is true
   α = Probability of TI error, aka the significance lv of the test
   typically: α = 0.05
  o TII error: occurs when H0 is not rejected when in fact it is false and should be rejected
   ß = TII error
   Typically: ß = 0.1 or 0.2
  o 1 – ß: Power of the test = the prob of rejection H0 when it is false
  o We like small α and large power
   If you want to reduce both types of error, we need to increase the sample size A LOT