Sampling
Randall Pruim
[email protected]
Calvin College
Sampling – p. 1/
What’s in the Bag?
Instructions.
1.
DO NOT look in the bag!
2. Shake the bag to mix its contents.
3. Reach in and draw out one milk lid. Record its color (green
or blue). Then put the lid back in the bag.
4. DO NOT look in the bag!
5. Repeat steps 2–4 until you have pulled out 10 lids and
recorded the color of each.
6. DO NOT look in the bag!
Sampling – p. 2/
What’s in the Bag?
Instructions.
1.
DO NOT look in the bag!
2. Shake the bag to mix its contents.
3. Reach in and draw out one milk lid. Record its color (green
or blue). Then put the lid back in the bag.
4. DO NOT look in the bag!
5. Repeat steps 2–4 until you have pulled out 10 lids and
recorded the color of each.
6. DO NOT look in the bag!
Question: What percentage of the lids in the bag are blue?
Sampling – p. 2/
What’s in the Bag?
Instructions.
1.
DO NOT look in the bag!
2. Shake the bag to mix its contents.
3. Reach in and draw out one milk lid. Record its color (green
or blue). Then put the lid back in the bag.
4. DO NOT look in the bag!
5. Repeat steps 2–4 until you have pulled out 10 lids and
recorded the color of each.
6. DO NOT look in the bag!
Question: What percentage of the lids in the bag are blue?
Bigger Question: What can we learn from this?
Sampling – p. 2/
Bigger is Better
Generally speaking we can be more sure of our estimates if the
are based on larger samples.
The mathematics of probability allows us to quantify this effect.
Sampling – p. 3/
Bigger is Better
Generally speaking we can be more sure of our estimates if the
are based on larger samples.
The mathematics of probability allows us to quantify this effect.
margin of error
sample size (95% confidence)
10
± 30%
milk lids sample
(Margins of error are somewhat smaller if the true percentage is very small or very large.)
Sampling – p. 3/
Bigger is Better
Generally speaking we can be more sure of our estimates if the
are based on larger samples.
The mathematics of probability allows us to quantify this effect.
margin of error
sample size (95% confidence)
10
100
± 30%
± 10%
pooling ten samples
(Margins of error are somewhat smaller if the true percentage is very small or very large.)
Sampling – p. 3/
Bigger is Better
Generally speaking we can be more sure of our estimates if the
are based on larger samples.
The mathematics of probability allows us to quantify this effect.
margin of error
sample size (95% confidence)
10
100
1000
± 30%
± 10%
± 3%
public opinion polls
(Margins of error are somewhat smaller if the true percentage is very small or very large.)
Sampling – p. 3/
Bigger is Better
Generally speaking we can be more sure of our estimates if the
are based on larger samples.
The mathematics of probability allows us to quantify this effect.
margin of error
sample size (95% confidence)
10
100
1000
10000
± 30%
± 10%
± 3%
± 1%
labor stats even better
(Margins of error are somewhat smaller if the true percentage is very small or very large.)
Sampling – p. 3/
Example 5.20
Consider a discrete rv X with the following pmf:
x
40 45 50
p(x) .2 .3 .5
E(X) = 40(.2) + 45(.3) + 50(.5) = 46.5
V (X) = 15.25
Sampling – p. 4/
Example 5.20
If we take a sample of size 2 (with replacement), there are nine
possible outcomes
x1
x2
p(x1 , x2 )
x̄
s2
40
40
40
45
45
45
50
50
50
40
45
50
40
45
50
40
45
50
.04
.06
.10
.06
.09
.15
.10
.15
.25
40
42.5
45
42.5
45
47.5
45
47.5
50
0
12.5
50
12.5
0
12.5
50
12.5
0
Sampling – p. 5/
Example 5.20
Consider a discrete rv X with the following pmf:
x
40 45 50
p(x) .2 .3 .5
The pmf’s for X̄ and S 2 are as follows:
40 42.5 45 47.5 50
pX̄ (x̄) .04 .12 .29 .30 .25
x̄
s2
0 12.5 50
pS 2 (s2 ) .38 .42 .20
E(X̄) = 40(.04) + 42.5(.12) + 45(.29) + 47.5(.3) + 50(.25) = 46.5
Sampling – p. 6/