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
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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…..
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What is the standard deviation of x-bar?
Assume that the standard deviation for the
population is 100.
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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)
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The confidence level, C, gives the probability that
the interval will capture the true value (μ) in
repeated samples (ex. 95% confidence interval)
PANIC
P: Parameter of interest- Define it
 A: Assumptions/conditions
 N: Name the interval
 I: Interval (confidence)
 C: Conclude in context

P: Parameters
Parameter: the statistical values of a
population (represented by a Greek letter)
 Define in first step of confidence interval
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µ= the true mean summer luggage weight for
Frontier Airline passengers
A: Assumptions
1.
2.
The data comes from an SRS from the
population of interest.
The sampling distribution of x bar is
approximately normal. (Normality).
1.
2.
3.
By central limit theorem if sample greater than 30
By graphing in your calculator if you have data
Individual observations are independent;
when sampling without replacement, the
population size N is at least 10 times the
sample size n. (Independence).
Assumptions: (in context of
problem)
1.
2.
3.
Given that the sample is random,
assuming it to be SRS.
Since n=100, the CLT ensures that the
sampling distribution is normally
distributed.
The population of frontier airline
passengers is certainly greater than
1000, (10x100) so the observations are
independent.
N: Name Interval

Interval: T-interval for means
I: Interval
Formula
 Plugged in from problem
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Interval: (179.03,186.97)
C: Conclude in Context
We are 95% confident that the true mean
summer luggage weight of Frontier Airline
passengers is between 179.03 pounds
and 186.97 pounds.
 We are __% confident that the true mean
[context] lies between (____,____).

Z*

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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?

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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.
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Example
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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
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Find the 95% Confidence Interval for 𝝁.
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What is the width of a 95% confidence interval for 𝝁?
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Oh, behave!
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How do Confidence Intervals behave?
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Let’s look at the margin of error portion of the
formula.

margin of error = z
n
*
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What happens when the sample size increases?
What about the Confidence Level increasing?
What happens when σ gets smaller?
Homework
Confidence Interval WS