Chapter 10
confidence intervals
Activity
Roll a Real die 50 times creating a Dot Plot to
keep track of your rolls
Calculate the sample mean of your 50 rolls, do
this on the calculator
What is the true mean and StDev of a real die?
What is an easy way to calculate this?
Is the mean 3.5?
 Construct a 95% confidence interval for
the true mean of the die.
 N: Name the Interval
 A: Assumptions/Conditions
 S: Stats from calculator
 C: Confidence Interval
 A: And
 R: Result in Context
95% Confidence Interval.
How many of these intervals captured μ,
which we know to be 3.5
Typically, about 1 student per class will not have
μ captured in their interval. This follows the
meaning of a 95% confidence interval.
What is the meaning of these intervals?
Meaning of a 95% C.I.
 The meaning is NOT:
 95% of all rolls are between(___) & (___).
 It is:
 If this process were to be done repeatedly,
about 95% of all intervals would capture the true
mean of the die.
In simplest terms:
If you did what you just did(roll 50 times, get the mean, make an
interval) like a million times, then 95% of those intervals you made
would capture μ.
90% Confidence Interval.
 Now on your calculator make a 90%
confidence interval for the same data
using Z*. See any differences?
 Try a 99% interval? What happens?
Write down on a piece of
paper the age of this man.
Write down your best guess.
I would say that I am 100%
confident that this man’s age is
between 0 and 110 years old.
But this is a useless
confidence interval.
If I want to me more accurate with my interval I have
to lower my confidence.
A more realistic interval might have a conclusion as…….
I am 90% confident that this man’s true age is between 26 and 38 years old.
Class of 2014
Class Estimates
Mean = 32.5
Med = 31
S = 9.997
A 95% CI gives (30.2, 34.8)
A 99% CI gives (29.4, 35.6)
Class of 2015
Class Estimates
This is Luke Wilson as Richie
Tenenbaum in the movie, “The
Royal Tenenbaums” from 2001.
He was 30 years old when this
movie was filmed.
What are some ways to
shrink your interval?
 Lower confidence.
 Higher sample size.
Confidence intervals—
Day 2
 Take your die—the one you made--and roll it
50 times.
 Create a Dot Plot
Use your calculator to calculate the mean and
StDev, make sure you use s for the StDev,
NOT σ
95% T Interval
Since we are using a die that you made, we must use a T
interval.
Why?
Because, the true standard deviation σ is unknown. We had
to calculate it. In real life and in most statistics that are not
made up, a T interval is used.
95% T Interval
conditions
We have a random sample of 50 rolls of a fake die.
Each roll is independent. Since our sample is more
than 30 our normal condition is met by the CLT.
x=
s=
(______, ______)
I am 95% confident that the true mean of my fake die is
between ____ & ____ because I used a method that captures
the true mean in 95 out of 100 attempts in repeated sampling.
Confidence intervals—
Proportions
 Roll your die 60 times to see the proportion
of 5s that you get.
 Write down the number of 5s that you get.
Lets create a 90% confidence interval for the proportion of 5’s.
90% Proportion Z interval
conditions
We have a sample of 60 independent rolls of a created fake
die. Our sample size of 60 rolls meets our Normality
requirement…. np
ˆ ³ 10 & n(1- pˆ ) ³ 10
60(__) = ___& 60(__) = ____
pˆ =
60
= .__
(______,_______)
I am 90% confident that the true proportion of 5’s that appear with my fake
die is between _____ & _______, because I used a method that captures the
true proportion in 90 out of every 100 attempts in repeated sampling.
How do we find the exact
sample size we want?
æs ö
Z ç
÷= m
è nø
*
æ p(1ˆ
ˆ ö
p)
Z *ç
÷= m
n ø
è
These are the
margin of error
formulas for Z
and T.
Back to the REAL Die.
How many times do we need to roll the die to have
our CI accurate to within ± .10 at 90% confidence?
æ
ö
s
*
Z ç ÷= m
è nø
After solving, n = 789.26
So we would need at least 790 rolls.
Use this formula
Z* = 1.645
s = 1.70783
m = .10
Margin of Error vs
Standard Error
æ s ö
Z ç
=
m
÷
è nø
*
Standard Error
With any of the margin of error formulas the standard deviation part is
called the standard error
Finding Z*
Z* is called the critical value, the common critical values
are 90,95, and 99. You should memorize these.
90 = 1.645
95 = 1.960
99 = 2.576
These can also be found on the table, see below
Finding Z*
Z* can also be found on all graphing calculators using
the invNorm function.
Desired confidence level
æ 1+ .90 ö
invNorm ç
è 2 ÷ø
Finding T*
T* depends on the sample size and the degrees of freedom(df)
df = n - 1
Example if we want 98% confidence with a sample size of 26,
what do we use for T*
Our df = 26 – 1
df = 25
Use the value where 98 and
25 meet
Finding T*
The TI-83 calculators do not have the invT function, the
TI-84 and TI-Nspire calculators do
Similar to invNorm but you need to also include the df
Finding Z* and T*
 In any case, you will always have the
tables with you. Fastest way is look at the
table.
What Critical t* Value
would you use?
 A 95% confidence interval based on n =
10 observations.
 A 99% confidence interval from an SRS
of 20 observations.
 An 80% confidence interval from a
sample of size 7.
How large a sample?
 A laboratory scale is repeatedly weighing
a 10 gram weight. The readings are
Normally distributed and the Standard
Deviation is known to be 0.0002 grams.
 How many measurements must be
averaged to get a margin of error of ±
0.0001 with 99% confidence?
How large a sample?
 Mr. Pines will be driving around Orange County
this weekend trying to estimate the true mean
gasoline price advertised at gas stations for the
Holiday season. Typically the standard
deviation this time of year for gas prices is σ =
.03
 How many gas stations must Mr. Pines
record prices for to have a margin of
error of ± 0.01 with 99% confidence?
Cutting the margin of error
 A very common question is how much
does your sample size have to increase
in order to cut the margin of error in half?
The sample needs to be 4x as large.
TRY IT ON
YOUR CALC
A sample size 4 times as large cuts the
margin of error in half
Reducing the margin of error
 Making the margin of error ½ as large we
had to multiply the sample size by 4.
 It follows that…..
 If the desired the margin of error is to be
1/n as big, then the sample size needs to
be multiplied by n2
Reducing the margin of error
 A poll taken at Rancho asked 45 students
whether they are in favor of school
uniforms. A confidence interval was
constructed.
 If they want to keep the same level of
confidence but divide the margin of error
in third, how many students will they need
to have in the poll?
405
The Meaning of a CI
 Explain what the meaning of a confidence
interval is.
You really need to work at this question.
Lets say you did a 95% confidence interval.
It means that if you took many samples and made a
confidence interval for each sample, then 95% of those
intervals would contain the true value.
The Meaning of a CI
 Explain what the meaning of a confidence
interval is.
You really need to work at this question.
Lets say you did a 95% confidence interval.
It DOES NOT MEAN that there is a 95% probability that
the true value is in this interval.
The Meaning of a CI
 Explain what the meaning of a confidence
interval is.
You really need to work at this question.
You need to beware when the word PROBABILITY is
attached to a CI.
The meaning of the % is how often your METHOD will
capture the true mean.
Review Problem
 In a recent survey of 1500 randomly
selected U.S. adults, 68% of the
respondents agreed with the statement “I
should exercise more than I do.”
 (a) Construct and interpret a 96% confidence
interval to estimate the proportion of the U.S.
adult population that would agree with the
statement.
 (b) For this study, state one source of potential
bias and how it would affect the estimate of the
proportion of adults who would agree with the
statement, “I should exercise more than I do.”
Ch 10 Confidence Intervals
MEANS
PROPORTIONS
Z interval or T interval
Z interval always used
Which interval should we
use?
Z interval for means, T interval for means, or Z interval for proportions
Sample was taken, they needed to know info about it, σ was unknown.
Which interval should we
use?
Z interval for means, T interval for means, or Z interval for proportions
Sample was taken, however σ was known.
Which interval should we
use?
Z interval for means, T interval for means, or Z interval for proportions
Proportion problem, only one option.
Which interval should we
use?
Z interval for means, T interval for means, or Z interval for proportions
Sample was taken, they needed to know info about it, σ was unknown.
Which interval should we
use?
Z interval for means, T interval for means, or Z interval for proportions
σ was known.
This is the formula for constructing a Z Interval by
hand, we will not be doing this. We will use the
graphing calculator only.
However, it is important to be familiar with it for a MC
type matching question.
This is the formula for constructing a T Interval by
hand, we will not be doing this. We will use the
graphing calculator only.
However, it is important to be familiar with it for a MC
type matching question.
This is the formula for constructing a Z Prop Interval
by hand, we will not be doing this. We will use the
graphing calculator only.
However, it is important to be familiar with it for a MC
type matching question.
Practice
Practice
Practice