| Quiz Ten, the Sampling Statistics
Quiz, will be due in SAKAI at 5
p.m. on Monday, April 7. It will include some statistical
calculations and some multiple choice items from Babbie, Chapter
Seven. You may take
this quiz as often as you like and the highest score will count.
If you wait until the last day to try this quiz, it is at your own risk
for any computer problems. |
The "margin of error" is a measure of how much our
sample statistic is likely to
vary from the population paramater. This based on
probability
theory. The larger your sample, the more certain your results,
and the smaller the margin of error. The
size of the population doesn't matter unless you are dealing with very
small populations. By convention, the "margin of
error" is two standard errors (twice the standard deviation of the
sampling distribution). This is explained in Babbie.
The size of the sample in practice depends on how many strats you want to sample. The margin of error for each stratum depends on the sample of individuals from that stratum..
To compute margins of error, follow the instructions in the Guide to Computing Margins of Error on the WEB site.
Some examples,
1. In a college class with 85 students, 32 of whom are
black, the mean on the midterm was 75. The standard deviation was 6.21.
What is the margin of error for this mean? This
is
a mean score question, so I use the formula
M = 2 * sd / SQRT(N)
;
M = 2 * 6.21/SQRT(85). = 1.35 points, NOT %%.
A n example using formula one. "A survey was
conducted with 1200 respondents in a state with 35,000,000, residents
of whom 25% are urban and 75% are rural." What is the margin of
error?
N = 1200. M = 1/sqrt(n). M =
1/34.64 M = .02886. Give the result as a percent with
one number after the decimal point. 2.9%
A survey was conducted with 1200 respondents in a state
with 35,000,000, residents of whom 25% are urban and 75% are
rural. Of these respondents, 55% preferred Hillary. We need
formula two because we have a percentage result for the sample, a
sample statistic. M = 2 * sqrt((p*(1-p))/N). To solve this
formula we need, N=1200. We need p, which is 55%, but we
make it .55. We work with proportions.
p=.55 q or 1-p=.45 p*(1-p) = .45
*.55 = .2475. Divide it by N and get .00020625
take the square root and get 0.01436 = .0287
as a proportion, which rounds of to 2.9%
A survey was conducted with 1200 respondents in a state
with
35,000,000, residents of whom 25% are urban and 75% are rural. Of
these respondents, 85% preferred Hillary. We need formula two
because
we have a percentage result for the sample, a sample statistic. =
.0206 or 2.1%
A sample has 400 respondents, what is the margin of
error? N=sqrt(400) = .05 or 5%
My county has 450,000 residents, how big of a sample do I
need to survey it? Not a good question, I have to ask, how big a
margin of error can you accept. I need 5% margin of error.
Use N = 1/(m*m). However, remember that M is the margin of
error expressed as a proportion. M = .05. N = 400.
I need to study the South Jersey area, consisten of five
counties (with various populations). I need a margin of error of
5% for each of the counties. How big a sample do I need? We
need 4009 for a 5% margin of error, but we need it for each of the
counties, so the sample must be 2000. This has to be drawn as 400
from each county.
9. A survey of the tri-county area has 356
respondents,
of whom 82 are black and 55 hispanic. What is the margin of error for
statistics
about the opinion of the hispanic residents? This
is a percentage question, but I am not given a statistic, a percentage
result. Use Formula one, M = 1/SQRT(N). What is N???
M - 1/sqrt(55). =
.1348.
This formula gives us a proportion, not a percent, or 13.5%.
Suppose
I said 61% of the hispanic respondents are voting for McGreevy.
That
is that statistic for the sample. The population paramater might
vary by as much as 13.5%. We could say that our "confidence
interval"
is between 61 - 13.5 and 61 + 13.5.
or between 47.5% and 74.5%. This means the election among
Hispanics
is "too close to call."
Suppose we had 400
Hispanics,
the margin of error would be 5%. For a sample of 1000, M =
1/SQRT(1000) or 1/31. or 3.2%
Suppose we wanted
a 5% margin of error, how large a sample do we need? 400.
Suppose
we want a 5% margin of error for each of five electoral districts, how
large a sample do we need? 5 * 400, or 2000.
Representative or random sample Chosen at random from either the total population (simple random sample) or from subgroups of the population.(stratified random sample)
In choosing a sample size, all that matters is the amount of error you can tolerate. The population size is not relevant.
A researcher wants to obtain a margin of error of no
more
than 2% in a survey of a county with a
population of 3,000,000. How large a sample is
needed?
N = 1/(M*M). M is the margin of error, expressed as a
proportion.
M = .02 because it says 2%. N = 1/(.02*.02) N=
1/.022 N =2500. Simple
random sample.
Suppose we were going to do this for five counties, and we wanted a 2% margin of error for each? How large a sample would we need? A 2% margin of error requires 2500, but we need it for eachcounty so we need 5 * 2500 or 12500. Stratified random sample, consisting of a simple random sample of each of the subgroups.
3. 59% of the respondents in a
survey
of a state with seven million Republican voters voted for Bush,
41% for Gore. There were 625
respondents.
What is the margin of error for the percent voting for Bush?
M = 2 * SQRT((p
* (1-p))/N). What is p, the proportion
of
respondents giving a certain response. The sample
statistic.
In this case, what is p? .59 What is N? M
=
2 * SQRT((.59 * (1-.59))/625). = .03935 as a proportion, or 3.94%
expressed as a percentage.
What does that mean?
We
can be "95% sure" that the population paramater (the true value for the
population) is witin 3.94% of the sample statistic. One way to
express
this is as a "confidence interval".
The lower bound of the
confidence
interval is the sample statistic minus the margin of error, in this
59-3.94
= 55.06%.
The upper bound of the
confidence
interval is the sample statistic plus the margin of error, in this case
59+3.94 = 62.94%.
We are confident that the true
figure, the "population paramater" is between 55.06% and 62.94%.