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.
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:

 critical
m  
 value
  standard deviation
  
  of the statistic



Confidence level
 Is
the success rate of the method
used to construct the interval
 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 µ is contained
in the confidence interval
 The probability that the interval
contains µ is 95%
 The method used to construct
the interval will produce
intervals that contain µ 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
  
x  z *

 n
estimate
Margin of error
Activity
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 of (context) lies
within the interval
_______ and ______.
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)
 known
 .2 
3.2  1.645
  3.0101, 3.3899
 3
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
 .2 
3.2  1.96
  2.9737, 3.4263
 3
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
 .2 
3.2  2.576
  2.9026,3.4974
 3
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 AHS
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
AHS students?
We are 95% confident that the true
mean SAT score for AHS 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 σ
= 105)
We are 95% confident that the true
mean SAT score for AHS 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:
  
m  z *

 n
Always round up to the nearest person!
The heights of AHS 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
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?
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

Graph examples of tcurves vs 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
How to find t*
Can also use invT on the calculator!
Use Table B for t distributions
t* value withlevel
5% isatabove
– &
Need
Lookupper
up confidence
bottom
95% is below
df on the so
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
Formula:
Standard
deviation of
Critical value
statistic
 s 
Confidence Interval : x  t * 

 n
estimate
Margin of error
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).
•  is unknown
 9.3 
114.9  2.056
  (111.22, 118.58)
 27 
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.
t-procedures can be used with some
skewness, as long as there are no
outliers.
 Larger n can have more skewness.

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.
(126.16, 189.56)
A diet guide claims that you will get 120
calories from a serving of vanilla
Note: confidence intervals tell us
yogurt. What does this evidence
if something is NOT EQUAL –
indicate?
never less or greater than!
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
 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