Download Ch 8 Notes - Campbell County Schools

Ch 8
Estimating with Confidence
Today’s Objectives
✓I can interpret a confidence level.
✓I can interpret a confidence interval in
context.
✓I can understand that a confidence
interval gives a range of plausible
values for the parameter.
The Basic Idea
• We will use the formulas and ideas from
the previous chapter.
• This time we will not know the true
population parameters. -- This is what
really happens.
• Inference is what Statistics is all about!
Vocabulary
•Point Estimator - what we’re using
•Point Estimate - the value of what we’re
using
•The best estimator has no bias and low
variability!
• Determine the point estimator you
would use and calculate the value of the
point estimate:
•
•
•
a) The makers of a new golf ball want to estimate the median distance
the new balls will travel when hit by a mechanical driver. They select a
random sample of 10 balls and measure the distance each ball
travels. Here are the distances (in yards):
285 266 284 285 282 284 287 290 288 285
b) The math department wants to know what proportion of its students
own a graphing calculator, so they take a random sample of 100
students and find that 28 own a graphing calculator.
How can we find the
confidence interval of our
mystery mean?
• Remember, if the population is Normal
then so is the sampling distribution.
x  
•and
x 

n
10% condition is met because we are sampling
from an infinite population in this case.
How can we find the
confidence interval of our
mystery mean?
• About how many standard deviation
from the mean represents a 95%
interval?
Therefore, we can estimate that the real mean lies
somewhere in this interval:
This happens in about 95% of all possible samples.
Confidence Interval for a
parameter:
estimate + margin of error
• Margin of Error: how close the estimate
tends to be to the unknown parameter
in repeated
• accounts for the variability due to
random selection. It does NOT
compensate for bias in data
collection!
• Two ways to write it
Confidence Level, C
• gives the overall success rate of the
method for calculating the confidence
interval.
• That is, in C% of all possible samples,
the method would yield an interval that
captures the true parameter value.
Confidence Interval
Visual
Interpreting Confidence Level and
Confidence interval
• Confidence Level: C% of all possible
samples of a given size from this
population will result in an interval that
captures the unknown parameter.
• Confidence Interval: I am C% confident
that the interval from __ to __ captures
the actual value of the [population
parameter in context]
• Don’t confuse confidence levels
with confidence intervals!
• Always interpret the interval,
only interpret the level when
asked
• Confidence level does NOT tell
us the chance that an interval
captures the population
parameter!
• It gives us a set of plausible
values
• What are the confidence level and
confidence interval of our mystery
mean?
• Also, interpret them.
Again, the confidence level is not a
probability!
The interval either does or does not capture
the true value of the population parameter.
According to www.gallup.com on
August 13, 2010, the 95% confidence
interval for the true proportion of
Americans who approved of the job
President Obama was doing was 0.44
+ 0.03
• Interpret the confidence level and
confidence interval.
Confidence Level and
the Length of the
Interval
Confidence Interval:
• statistic + (critical value) (standard deviation of
statistic)
(this is on your formula sheet)
The critical value depends on both, the confidence
level, C, and the sampling distribution of the statistic.
How to get a small
margin of error?
1. Lower your confidence level
2. Increase your sample size
WHY?
Can you think of pros and cons of
each?
Conditions to check before
calculating a confidence interval:
1. Random
2. Normal
3. Independent
Read page 479 about these in detail!
Random
• Randomization must be used correctly!
• If not, our estimates might be biased
and we shouldn’t have any confidence
that the intervals we calculate will
actually contain the value of the
parameter we’re trying to estimate.
Normal
• all calculations depend on the fact that
the sampling distribution is
approximately Normal!
• We will be over confident is we assume
Normality when the population is in
reality skewed.
• Remember there are two different tests
for Normality and it depends on if we
are using sample means or sample
proportions!
Independent
• our standard deviation formulas depend
on replacing the sample; rarely is that
the case.
• This makes checking the 10% condition
more critical!
Why these 3 conditions?
• They each are related to the 3 parts of
the formula:
• Random ensures the statistic is
unbiased.
• Normal ensures that we are using the
correct critical value
• Independence ensures we are using the
correct formula for the standard
deviation of the statistic