•
This is the probability that the estimate procedure will generate an interval
that does not contain
•
The confidence coefficient (1- α) is the probability that it will contain
(.95 for
example)
•
Confidence level = 100(1- α) %
(aka 95%)
•
11

Confidence
•
In order to figure out z-scores to use for the confidence
intervals, we use the z-score at α/2
•
We take a confidence we want, find z by doing 1-α/2
12

Example
•
For a 95% interval, α/2 = .05/2 = 0.025 so I look up 1-.025 = .
975 on the table to see what’s to the left of that.
I get z=
1.96
•
For a 90% interval, α/2 = .1/2 = .05 so I look up 1-.05 = .95
on the table and see z = 1.645
•
What about 99%?
13

For all problems
•
Figure out the z-score from the confidence level and use it in
the formula:
- z/
<= <= + z/
Margin of error = z/
For a given confidence level, the larger the population standard
deviation, the wider the confidence level.
•
14

Example
•
Let the sd of a population be 0.05 in the last example instead
of 0.03
Compute the 95% confidence interval based on the
sample information.
•
Our mean was 1.02 and sample size was 25
•
- z/
<= <= + z/
•
15

Example
•
Let the sd of a population be 0.05 in the last example instead of 0.03
Compute the 95% confidence interval based on the sample information.
•
Our mean was 1.02 and sample size was 25
•
- z/
<= <= + z/
•
1.02 +/- 1.96(.05/
)
•
1.02 +/- 0.02 = 1 to 1.04
•
The width is 04 here (.02 X 2)
•
16

Other Rules
•
For a given confidence level and population standard
deviation, the smaller sample n, the wider the confidence
interval.
•
For a given sample size n and population standard deviation,
the greater the confidence level, the wider the confidence
interval.
17

Example
•
Compute the 99% confidence interval instead from the last
example.
•
sd =0.03, mean= 1.02, n=25
•
- z/
<= <= + z/
•
18

Example
•
Compute the 99% confidence interval instead from the last
example.
•
sd =0.03, mean= 1.02, n=25
•
- z/
<= <= + z/
•
1.02 +/- 2.576 (/
= 1.02 +/- .015 = 1.005 to 1.035
•
19

Confidence interval for pop mean
when sigma is unknown
•
In order to use a confidence interval for a population mean,
Xbar needs to be normally distributed.
•
Another standardized statistic T is used which uses S as a
replacement for sigma when the population standard
deviation is not known.
•
T =
•
This is called a Student’s distribution, or
a t distribution
•
20

T-distribution
•
If a random sample of size n is taken from a normal distribution with a finite
variance, then the statistic T =
follows the T distribution with (n-1) degrees of
freedom.
•
The T distribution is actually a family of distributions which are similar to x in that
they are bell shaped and symmetric around 0. However, T distributions all have
slightly broader tails.
•
The degrees of freedom define the broadness of the tails of the distribution. The
fewer the “df” the broader the tails.
•
They are also the number of independent pieces of information that go into the
calculation of a given statistic and can be “freely chosen.”
•
21

T table
•
Just like the normal table, there is a T-table on Table 2.

#### You've reached the end of your free preview.

Want to read all 40 pages?

- Fall '18
- Tella
- Normal Distribution, Jared Beane