Download STAB22 Statistics I

STAB22 Statistics I
Lecture 19
1
Sampling Distributions

Suppose you do random sampling. How
do you answer questions like:




How accurate are sampling results?
What is a good sample size n?
Is it “reasonable” to assume Canadians
spend on average 2 hours watching TV?
Use Sampling Distributions
2
Population
Population parameter
(TV watching times)
A
5
B
C
D
E
3
2
2
3
2
5
3
2
Sampling Error
(sample mean)
(n=2)
3
53 2 23

3
5
Sample Statistic
Sampling
5
(population mean)
2
2
3
33
X
3
2
X    33  0
3
53
X
4
2
X    43 1
Sample statistic takes different values w/ different
probabilities → X follows a Sampling Distribution
3
Sampling Distribution
Population (N=5)
A
B
C
D
Sampling (n=2)
E
Sample
5
3
2
2
3
Sampling Distribution
Prob
.4
.3
.2
.1
2
2.5
3
3.5
4
X
● 10% of samples with error = 0
● 70% of samples with error ≤ 0.5
X
A,B
4
A,C
3.5
A,D
3.5
A,E
4
B,C
2.5
B,D
2.5
B,E
3
C,D
2
C,E
2.5
D,E
2.5
4
Mean and Variance of
Sampling Distribution

What is sampling distribution’s mean?
EX   
 is population mean


0.4
0.3
0.2
0.1
2
2.5
3
3.5
4
What is sampling distribution’s variance?
2


Var  X  
, SD ( X ) 
n
n
 2 is population variance
Sample
size n=5
Sample
size n=10
5
Sampling Distribution Shape
For large n (typically n≥30), sampling distribution
of mean approaches Normal, for any population!!
6
Central Limit Theorem

For population w/ mean µ, variance σ², if


Samples are random and independent
Sample size n is large enough (typically n≥30)

If sampling without replacement, n must be less than
10% of population
Sampling distribution of X is approximately
2
Normal, with mean  and variance  / n

But we anyways don’t know µ or σ² !!!
Is the Central Limit Theorem useful?
7
Example

Estimate average Canadian TV-watching time
Sample n=100 people; what is the probability your
sample mean has sampling error < 0.2 hrs?
(assume σ = 2hrs)
8
Example

What is the probability your sample mean has
sampling error ≤ 0.2 hrs, if sample size is n=10,000?
9
Example



Bottling machine fills bottles with amount of soda that
is Normally distributed with µ=500ml & σ=25ml
Quality control collects random sample of 25 bottles.
If mean is <490ml or >505ml, machine is serviced.
Find the probability that the machine is serviced,
even if it is working fine.
10
11
Sample Proportion

Bernoulli population: Each population subject
belongs to one of two categories


Population proportion (parameter) p


E.g. proportion of male students
Random sample of size n:



E.g. male/female, employed/unemployed
X = # of sample subjects in category of interest
X follows Binomial with parameters n, p
Sample Proportion: pˆ  X / n

Sample estimate of population proportion p
12
Sampling Distribution of
Sample Proportion

For population w/ proportion of success p, if


Samples are random and independent
Sample size n is large enough, so that

np  10

n(1  p )  10
If sampling without replacement, n must be less than
10% of population
Sampling distribution of pˆ is approximately
Normal, with mean p and variance p(1  p) / n
E  pˆ   p
p (1  p)
Var  pˆ  
 SD ( pˆ ) 
n
p (1  p )
n
13
Example


University students will vote on proposal, for which
p=60% are in favor. You make a small poll, asking
n=25 students for their opinion.
What is the probability that the poll gives you wrong
results (i.e. rejects the proposal)?
14