Download Confidence Intervals

Confidence
Intervals with
Means
Rate your confidence
0 - 100
Name my age within 10 years?

within 5 years?

within 1 year?

Shooting a basketball at a wading pool,
will make basket?
 Shooting the ball at a large trash can, will
make basket?
 Shooting the ball at a carnival, will make
basket?

What happens to your
confidence as the interval
gets smaller?
The larger your confidence,
the wider the interval.
Guess the number
Teacher will have pre-entered a number
into the memory of the calculator.
 Then, using the random number generator
from a normal distribution, a sample mean
will be generated.
 Can you determine the true number?

Point Estimate
 Use
a single statistic based on
sample data to estimate a
population parameter
 Simplest approach
 But not always very precise due to
variation in the sampling
distribution
Confidence intervals
 Are
used to estimate the
unknown population mean
 Formula:
estimate + margin of error
Margin of error
Shows how accurate we believe our
estimate is
 The smaller the margin of error, the
more precise our estimate of the true
parameter
 Formula:

Confidence level
 Is
the success rate of the method
used to construct an interval that
contains that true mean
 Using this method, ____% of the
time the intervals constructed
will contain the true population
parameter
What does it mean to be
95% confident?
 95%
chance that m is contained
in the confidence interval
 The probability that the interval
contains m is 95%
 The method used to construct
the interval will produce
intervals that contain m 95% of
the time.
Critical value (z*)
Found from the confidence level
 The upper z-score with probability p
lying to its right under the standard
normal curve

z*=1.645
z*=1.96
z*=2.576z*
Confidence level tail area
90%
95%
99%
.05
.025
.005
1.645
.05
.025 1.96
.005
2.576
Confidence interval for a
population mean:
Standard
Critical
value
deviation of
the statistic
estimate
Margin of error
Steps for doing a confidence
interval:
Assumptions –
• SRS from population
• Sampling distribution is normal (or
approximately normal)
 Given (normal)
 Large sample size (approximately
normal)
 Graph data (approximately normal)
• σ is known
2) Calculate the interval
3) Write a statement about the interval in
the context of the problem.
1)
Statement: (memorize!!)
We are __________%
confident that the true
mean context lies
within the interval
_______ and ______.
Confidence Interval Applet

http://bcs.whfreeman.com/tps4e/#62864
4__666391__

The purpose of this applet is to
understand how the intervals move but
the population mean doesn’t.
A test for the level of potassium in the blood is
not perfectly precise. Suppose that repeated
measurements for the same person on different
days vary normally with σ = 0.2. A random
sample of three has a mean of 3.2. What is a
90% confidence interval for the mean potassium
level?
Assumptions:
Have an SRS of blood measurements
Potassium level is normally distributed (given)
s known
We are 90% confident that the true mean
potassium level is between 3.01 and 3.39.
95% confidence interval?
Assumptions:
Have an SRS of blood measurements
Potassium level is normally distributed
(given)
s known
We are 95% confident that the true mean
potassium level is between 2.97 and 3.43.
99% confidence interval?
Assumptions:
Have an SRS of blood measurements
Potassium level is normally distributed
(given)
s known
We are 99% confident that the true mean
potassium level is between 2.90 and 3.50.
What happens to the interval as the
confidence level increases?
the interval gets wider as the
confidence level increases
How can you make the margin
of error smaller?

z* smaller
(lower confidence level)

σ smaller
(less variation in the population)

Really cannot
n larger
change!
(to cut the margin of
error in half,
n must be 4 times as big)
A random sample of 50 CHS
students was taken and their mean
SAT score was 1250. (Assume σ =
105) What is a 95% confidence
interval for the mean SAT scores of
CHS students?
We are 95% confident that the true
mean SAT score for CHS students is
between 1220.9 and 1279.1
Suppose that we have this
random sample of SAT scores:
950 1130 1260 1090 1310 1420 1190
What is a 95% confidence interval for
the true mean SAT score? (Assume s
= 105)
We are 95% confident that the true
mean SAT score for CHS students is
between 1115.1 and 1270.6.
Find a sample size:

If a certain margin of error is
wanted, then to find the sample size
necessary for that margin of error
use:
Always round up to the nearest person!
The heights of CHS male students
is normally distributed with σ =
2.5 inches. How large a sample is
necessary to be accurate within
+/- .75 inches with a 95%
confidence interval?
n = 43
Student t-Distribution
In a randomized comparative experiment on
the effects of calcium on blood pressure,
researchers divided 54 healthy, white males
at random into two groups, takes calcium or
placebo. The paper reports a mean seated
systolic blood pressure of 114.9 with
standard deviation of 9.3 for the placebo
group. Assume systolic blood pressure is
normally distributed.
Can you find a z-interval for this problem?
Why or why not?
No, don’t know σ
Only sample s
Gossett Story
Statistics are
Parameters are
variables – each
constant – don’t
sample s will
expect the shape
 to
Can
you use
s insteadcause
of σ when
the shape
change,
just
calculating
z-score (so
that youaway
can find
to change
shift basedaon
the
+/- 3 σin)?
from a normal
changes
ẋ
distribution
 Not exactly. Look at the
two equations.
ẋ has a normal
distribution

Do t-score bingo.
Student’s t- distribution
Developed by William Gosset
 Continuous distribution
 Unimodal, symmetrical, bell-shaped
density curve
 Above the horizontal axis
 Area under the curve equals 1
 Based on degrees of freedom

T-Curves
Basic properties of t-Curves
 Property 1: The total area under a t-curve
equals 1.
 Property 2: A t-curve extends indefinitely
in both directions, approaching the
horizontal axis asymptotically
 Property 3: A t-curve is symmetric about
0.
T-curves continued

Property 4: As the number of degrees of
freedom becomes larger, t-curves look
increasingly like the standard normal curve
How does t compare to
normal?
 Shorter
& more spread out
 More area under the tails
 As n increases, t-distributions
become more like a standard
normal distribution
Graph examples of
t- curve vs normal curve
z – Score and t - Score
Confidence Interval
Formula:
Standard
deviation of
Critical value
statistic
estimate
Margin of error
How
toalso
find
Margin
when
Can
use
invT onof
theerror
calculator!
Fornot
90%available
confidence–level,
σ is
find5%
t*is above



and 5% is below
Use Table B for t distributions
Need upper t* value with 5% above – so
Look up confidence level at bottom &
0.95
is
p
value
degrees of freedom (df) on the sides
df = n – 1
invT(p,df)
Find these t*
90% confidence when n = 5 t* =2.132
95% confidence when n = 15 t* =2.145
Assumptions for t-inference
 Have
an SRS from population
 σ unknown
 Normal distribution
–Given
–Large sample size
–Check graph of data
For the Ex. 4: Find a 95% confidence
interval for the true mean systolic
blood pressure of the placebo group.
Assumptions:
• Have an SRS of healthy, white males
• Systolic blood pressure is normally distributed
(given).
• s is unknown
We are 95% confident that the true mean systolic
blood pressure is between 111.22 and 118.58.
Robust
An inference procedure is ROBUST if
the confidence level or p-value doesn’t
change much if the assumptions are
violated.
 For adequately sized samples (n≥30) ,
t-procedures can be used with some
skewness, as long as there are no
outliers.

Ex. 5 – A medical researcher measured
the pulse rate of a random sample of 20
adults and found a mean pulse rate of
72.69 beats per minute with a standard
deviation of 3.86 beats per minute.
Assume pulse rate is normally
distributed. Compute a 95% confidence
interval for the true mean pulse rates of
adults.
(70.883, 74.497)
Another medical researcher claims
that the true mean pulse rate for
adults is 72 beats per minute. Does
the evidence support or refute this?
Explain.
The 95% confidence interval contains
the claim of 72 beats per minute.
Therefore, there is no evidence to doubt
the claim.
Ex. 6 – Consumer Reports tested 14
randomly selected brands of vanilla
yogurt and found the following
numbers of calories per serving:
160 200 220 230 120 180 140
130 170 190 80 120 100 170
Compute a 98% confidence interval for
the average calorie content per serving
of vanilla yogurt.
Boxplot shows approx normal distr.
(126.16, 189.56)
Note: confidence intervals tell us
A dietifguide
claims
that you
will–get 120
something
is NOT
EQUAL
calories
fromless
a serving
of vanilla
never
or greater
than!
yogurt. What does this evidence
indicate?
Since 120 calories is not contained
within the 98% confidence interval, the
evidence suggest that the average
calories per serving does not equal 120
calories.
Some Cautions:
 The
data MUST be a SRS from the
population (must be random)
 The formula is not correct for more
complex sampling designs, i.e.,
stratified, etc.
 No way to correct for bias in data
Cautions continued:
 Outliers
can have a large effect
on confidence interval
know σ to do a z-interval
– which is unrealistic in
practice
 Must