Transcript Section 10.1 Second Day - Cabarrus County Schools

Section 8.1
Confidence Intervals for a
Population Mean
What has been the point of all this?
The point – Inference!
 The whole point of statistics is so we can
“infer” information about our population
from our sample data.
 Statistical inference – methods for drawing
conclusions about a population from
sample data

Confidence Intervals

Allow you to estimate the value of a
population parameter.
Let’s start off easy

Suppose you want to estimate the mean
SAT Math score for the more than 350,000
high school seniors in California who take
the SAT. You look at a simple random
sample of 500 California high school
seniors who took the SAT. The mean of
your sample is x-bar = 461. What can we
say about μ for the population of California
seniors?
What can we come up with?

We are looking at sample means, so the
distribution is approximately…..



What is the standard deviation of x-bar?
Assume that the standard deviation for the
population is 100.


Normal by…
CLT
4.5
We know that x-bar should be fairly close to μ. If
we want to know with 95% confidence, then we
want to know what values μ could be between
so that there is 95% area between. Does the
95% sound familiar?
Confidence Interval

Confidence interval = estimate ± margin of error

Estimate is the x-bar in this example (what you
get from your sample)
Margin of error is how far we are willing to go
from the estimate (9 in the last example)


The confidence level, C, gives the probability that
the interval will capture the true value (μ) in
repeated samples (ex. 95% confidence interval)
Steps to Construct a
Confidence Interval




1) Identify the population of interest and the
parameter (μ, p, or σ) you want to draw
conclusions about.
2) Verify the conditions for the selected
confidence interval.
3) If the conditions are met, use the formula for
a CI:
 CI = estimate ± margin of error
4) Interpret your results in the context of the
problem.
Step 1: Population of Interest
and Parameters
In this section, our parameter of interest is
a population mean, or μ.
 We will use x-bar as our estimate of μ.
 Find x-bar if it is not given to you.

Step 2: Conditions

There are conditions that MUST be satisfied before you construct a
confidence interval.


The data must be a SRS from the population of
interest.
 They will tell you this in the question.
2) The sampling distribution of x-bar is approximately
normal.
 If we’re given the actual data, we can plot it to see
if the distribution is approximately symmetric
(remember normal probability plots).
 Otherwise, we can assume that the distribution is
normal if n is large (CLT).
 The sampling distribution is approximately normal
if the population distribution is approximately
normal
1)
Step 3: The Formula


The general formula for a Confidence Interval is
CI = estimate ± margin of error.
Specifically, the CI formula for μ is
Our unbiased
estimator of μ.

xz
n
*
Z* will vary based
on the Confidence
Level.
This is the
standard
deviation of the
sampling
distribution of xbar.
Z*


We know that for our interval to have 95%
confidence, we should go out 2 standard
deviations from x-bar.
What about levels of confidence other than 6895-99.7?




Draw a picture.
Shade the middle region.
Find the area TO THE LEFT of z*. Look this value up
in the BODY of Table A. Find the Z score that
corresponds to that area.
OR
You can check for common z* upper tail values by
looking at the bottom of Table C.
Common Confidence Levels
Confidence
Level
Tail Area
90%
.05
95%
99%
Z*
Step 4: Express your results in
CONTEXT
Fill in the blanks….
 We are (insert confidence level) confident
that the true (mean or other parameter) of
(put in your context) is between (lower
bound) and (upper bound).
 If you forget this, you can find it at the end
of your book.

Example



Here are measurements (in mm) of a critical
dimension on a sample of auto engine
crankshafts: 224.120, 224.001, 224.017,
223.982, 223.989, 223.961, 223.960, 224.089,
223.987, 223.976, 223.902, 223.980, 224.098,
224.057, 223.913, 223.999
The data come from a production process that is
known to have standard deviation σ = 0.060mm.
The process mean is supposed to be μ = 224
mm but can drift away from this target during
production.
Give a 95% confidence interval for the process
mean at the time these crankshafts were
produced.
You try this one

A hardware manufacturer produces bolts used to assemble
various machines. Assume that the diameter of bolts produced
by this manufacturer has an unknown population mean 𝝁 and
the standard deviation is 0.1 mm. Suppose the average
diameter of a simple random sample of 50 bolts is 5.11 mm.

Calculate the margin of error of a 95% confidence interval for 𝝁

Find the 95% Confidence Interval for 𝝁.

What is the width of a 95% confidence interval for 𝝁?


Oh, behave!

How do Confidence Intervals behave?

Let’s look at the margin of error portion of the
formula.

margin of error = z
n
*



What happens when the sample size increases?
What about the Confidence Level increasing?
What happens when σ gets smaller?
Homework
Confidence Interval WS