Download Section 8.1 Estimating a Population Mean σ

Section 8.1 Estimating a Population Mean
How do we estimate a population mean? We take a random sample of
a reasonable size.
Earlier on we discussed parameters which were values of the
population such as  and  . Statistics were values of a sample and this
included x and s . A new term that involves parameters and statistics is the
point estimate.
Point estimate: is a value of a statistic that is used as an estimate of a
parameter. So x would be an estimate of  and s would be an estimate of
.
Example
Find an estimate for 
The number of offspring from the Eastern Cotton Mouth snakes is believed
to have decreased due to human encroachment. So 44 female snakes were
randomly sampled and the number of offspring from each snake was
counted. The sample was
5
12
7
7
6
8
12
9
7
4
9
6
12
7
5
6
10
3
10
8
8
12
5
6
10
11
3
8
4
5
7
6
11
7
6
8
8
14
8
7
11
7
5
6
 x  334
  2.4 young/litter
x  7.59 young/litter
So x is a point estimate for the mean number of young per litter,
eastern cotton mouth snakes.
A question to ask is our point estimate
x
likely to be equal to
 of all
 exactly?
If x is not going to be exactly the same as the true mean  is there a
range of values that  could be between? Yes that would be a confidence
interval
Confidence Interval (CI): is an interval of numbers obtained from a point
estimate of a parameter
How to we determine how sure or confident that
We have a confidence level
 is within that interval?
Confidence Level: The confidence we have that the parameter lies in the
confidence interval (i.e. that the confidence interval contains the parameter.
If we are two standard deviations away from the mean  using our x then
we would be 95.44% confident that the mean number of young/litter for our
snakes would be within these limits:
x  2

which would result in
n
7.59  2 
2.4
44
So the lower limit would be 6.87 young/litter and the upper limit would be
8.31 young/litter.
Confidence-interval estimate: the confidence level and confidence interval
So for our example we would be 95.44% confident that the mean number of
offspring for the Eastern Cotton Mouth snakes would be between 6.87
young/litter and 8.31 young/litter.
Does this mean anything else?
It means that if we obtained 20 groups with 44 pregnant female Eastern
Cotton Mouth snakes and set up a confidence interval for each one, then we
would expect 1 out of the 20 (5%) not to contain the true mean.
This situation is best represented by a drawing that will be shown in class.
Note that each Confidence Interval (CI) may or may not contain the true
population parameter.
The probability of any particular interval containing the true population
parameter is either 1 or 0. 1 means that it does contain the true parameter.
O means that it does not contain the true parameter.
If the confidence interval is 90% instead of 95%, then two out of the twenty
confidence intervals or (10 %) would not contain the true population
parameter.