# Point and Interval Estimation with Confidence

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```					Point and Interval Estimation
with Confidence

Another Spectacular
Power Point
Presentation
Point Estimates
• Definition:
A single number that is based on
sample data, and represents a reasonable
value of the characteristic of interest.
- Things Like:
x    s     M
- The simplest approach to estimating a
population value.
Lets Look at an Example
•   A recent survey asked     • What could we use to
high school senior how        estimate µ?
many hours they spent     •   M or x
on line each day. Find    •   Find both:
the point estimate of µ
with the data below.      •   x = 46.25 = 4.625
4 7.25 5 5.25 6.25                  10
6.25 4.5 3 .5 4.25            M = 4.75
•   Which one is better?
•   Recall when it is best to
use each one…
Confidence Intervals
(estimating µ)
• Things to Remember About the Distribution of x:
1) Has a normal distribution (by CLT)
2) Is an unbiased estimator of µ (x = µ)
3) If permission slip is granted:
s = σ/√n
•   Issues with Point Estimates: if we re – do the
sample, will we get same stat?
Confidence Intervals
Def: the interval of reasonable values of the
characteristic of interest
Pattern for the formula: (what’s on AP test)
statistic ± critical value(standard deviation of stat)

For Estimating µ: x ± z*(σ/√n)

Margin of Error
Confidence Levels
Def: the success rate of the method used to
construct the interval.

Ex: A 95% confidence interval = if we took lots of samples
and found the mean over and over again, in the long run
95% of the resulting intervals would capture the true
value of the parameter.

If we go through the same process with the same sample
size, σ, and z*, but different x, 95% of those intervals
will contain µ. (See pg. 511)
Calculating Confidence Intervals
What is z*?
A location on the normal curve that
results in the needed confidence level.

Ex:                   95%
-z*          z*

Use Table C!
z* = 1.96
Calculating Confidence Intervals
Let’s try another z*:

98%
-z*         z*

What is z*?
z* = 2.326
Critical Values
z* is also called a
critical value. Any
ideas why?

The area outside of the
interval are called
critical areas
Calculating Confidence Intervals
Suppose we need to verify the amount of
active ingredient in a new drug. σ is
known to be .0068 grams per liter. Here
are the results of 3 measurements. Find a
99% confidence interval for the true
concentration.
.8403     .8363      .8447
Calculating Confidence Intervals

• What we Need:
x         z*        σ         n
.8404 2.576         .0068      3
• Fill in the formula: x ± z*(σ/√n)
.8404 ± 2.576(.0068/√3)
.8404 ± .0101
.8303 to .8505
Calculating Confidence Intervals

• Use the same problem, but do a 90% C.I.
.8404 ± 1.645 (.0068/√3)
.8404 ± .0065
.8339 to .8469
• Compare the 90% to 99%
What happened to the area? z*? M of E?
See why we are “more” confident!
How Intervals Behave
1) Higher confidence gives up “precision.”
2) Want to have a small margin of error:
- smaller z*: again calls for lower
confidence
- smaller σ: smaller variation among
individuals
- larger n: better picture of population
AKA: Law of Large Numbers
Need 4x sample size to cut m of e in half
Choosing a Sample Size
Usually decided by desired margin of error
(m)

Formula: m ≥ z*(σ/√n)

• Make sure to round appropriately!!
• Population still needs to be 10X as big as
sample!
Choosing a Sample Size
• Example: From previous problem, what
sample size is required to produce results
±.005 with a C.I. of 95%?

.005 ≥ 1.96 (.0068/√n)

Do some algebra! √n≥(1.96∙.0068)/.005
n ≥ 7.1 so n = 8 because 7 is not enough!
Cautions!
1) Data must come from a SRS! We don’t
get into stuff for stratified, or blocked.
2) Bias is still bad! Fancy formulas cannot