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Confidence
Intervals
Chapter 9
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 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
Confidence level
90%
95%
99%
z*=1.645
tail area z*=1.96
z*=2.576z*
.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 z-interval
for means:
1) Assumptions –
•
•
•
•
SRS from population
Sample is < 10% of the population
Independence among data values is plausible
Sampling distribution is normal (or approximately
normal)
•
•
•
•
Given (normal)
Large sample size (n>30)
Graph data (unimodal and relatively symmetric)
 is known
2) Calculate the interval
3) Write a conclusion about the interval in the
context of the problem.
Conclusion:
We are ________% confident
that the true mean context lies
within the interval ______ and
______.
•
•
The NAEP (National Assessment of
Educational Progress) includes a short
test of quantitative skills, covering basic
arithmetic and the ability to apply it. The
standard deviation of the test is 60.
Suppose a random sample of 50 young
adult men are taken from a large
population. If the sample mean of their
scores is 265, what is a 95% confidence
interval for the true mean score for young
adult men on this test?
What about a 90% confidence interval?
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
 = 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 
  3.0101, 3.3899 
3.2  1.645
 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)
 known
 .2 
  2.9737, 3.4263
3.2  1.96
 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)
 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)
•
 smaller
(less variation in the population)
• n larger
Really cannot
(to cut the margin of error
in half, n must
change!
be 4 times as big)
A random sample of 50 PWSH
students was taken and their mean
SAT score was 1250. (Assume  =
105) What is a 95% confidence
interval for the mean SAT scores of
PWSH students?
We are 95% confident that the true
mean SAT score for PWSH 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  = 105)
We are 95% confident that the true
mean SAT score for PWSH 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 PWSH male students
is normally distributed with  = 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
t- curves vs normal curve
Comparison of normal and t distibutions
df = 2
df = 5
df = 10
df = 25
Normal
-4
-3
-2
-1
0
1
2
3
4
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 for t distributions
t* value with
5% at
is above
• Need
Lookupper
up confidence
level
bottom– &
so 95% is below
df on the sides
• df = n – 1
invT(p,df)
Find these t*
90% confidence when n = 5
95% confidence when n = 15
t* =2.132
t* =2.145
Formula:
Standard
deviation of
Critical value
statistic
Confidence Interval :
 s 

x  t * 
 n
estimate
Margin of error
Assumptions for t-inference
 unknown
Have an SRS from population
Sample is < 10% of the population
Independence among data values is
plausible
• Sampling distribution is normal (or
approximately normal.
•
•
•
•
– Given (population normal)
– Graph data (unimodal and relatively symmetric
with no outliers) or large sample size
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
• 27 white males (placebo group) is <10% of white males
• We assume blood pressures are independent
• Systolic blood pressure is normally distributed (given).
•  is unknown, so we will construct a t-interval
s
x t
, df  26
n
 9.3 
114.9  2.056
  (111.22, 118.58)
 27 
*
We are 95% confident that the true mean systolic blood pressure of
healthy white males 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
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
• 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
• Must know  to do a z-interval –
which is unrealistic in practice