# Ch10 Confidence Intervals

```Rate your confidence
0 - 100
0 Name my age within 10 years?
0 within 5 years?
0 within 1 year?
0 Shooting a basketball at a wading pool, will
0 Shooting the ball at a large trash can, will
0 Shooting the ball at a carnival, will make
What happens to your
confidence as the interval
gets smaller?
The larger your confidence,
the wider the interval.
Simulation
Point Estimate
0Use a single statistic based on
sample data to estimate a
population parameter
0Simplest approach
0But not always very precise due to
variation in the sampling
distribution
Confidence intervals
0Are used to estimate the
unknown population mean
0Formula:
estimate + margin of error
Margin of error
0 Shows how accurate we believe our estimate is
0 The smaller the margin of error, the more precise our
estimate of the true parameter
0 Formula:
 critical
m  
 value
  standard deviation
  
  of the statistic



Confidence level
0Is the success rate of the method
used to construct the interval
0Using this method, ____% of the
time the intervals constructed
will contain the true population
parameter
Critical value (z*)
0 Found from the confidence level
0 The upper z-score with probability p lying to
its right under the standard normal curve
z*=1.645
z*=1.96
z*=2.576
Confidence level tail area
z*
.05
1.645 .05
90%
.025
95%
99%
.025
.005
1.96
2.576
.005
What does it mean to be 95%
confident?
095% chance that m is contained in
the confidence interval
0The probability that the interval
contains m is 95%
0The method used to construct the
interval will produce intervals that
contain m 95% of the time.
Confidence interval for a
Standard
populationCritical
mean:
deviation of the
value
statistic
  
x  z *

n

estimate
Margin of error
Steps for doing a confidence
interval:
1) Assumptions –
•
•
SRS from population
Sampling distribution is normal (or approximately
normal)
0
0
0
•
Given (normal)
Large sample size (approximately normal)
Graph data (approximately normal)
 is known
2) Calculate the interval
3) Write a statement about the interval in the
context of the problem.
Statement: (memorize!!)
We are ________% confident
that the true mean 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
 = 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
Critical value (z*)
0 Found from the confidence level
0 The upper z-score with probability p lying to
its right under the standard normal curve
z*=1.645
z*=1.96
z*=2.576
Confidence level tail area
z*
.05
1.645 .05
90%
.025
95%
99%
.025
.005
1.96
2.576
.005
How can you make the margin of
error smaller?
0 z* smaller
(lower confidence level)
0  smaller
(less variation in the population)
Really cannot
0 n larger
change!
(to cut the margin of error
in half, n must
be 4 times as big)
A random sample of 50 BGHS
students was taken and their mean
SAT score was 1250. (Assume  =
105) What is a 95% confidence
interval for the mean SAT scores of
BGHS students?
We are 95% confident that the true
mean SAT score for BGHS students
is between 1220.9 and 1279.1
Find a sample size:
0 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 BGHS 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
Homework pg. 632-633 711, 13, 15
t- distribution
0 Developed by William Gosset
0 Continuous distribution
0 Unimodal, symmetrical, bell-shaped density curve
0 Above the horizontal axis
0 Area under the curve equals 1
0 Based on degrees of freedom
Graph examples of t- curves vs normal curve
How does t compare to
normal?
0Shorter &amp; more spread out
0More area under the tails
0As n increases, t-distributions
become more like a standard
normal distribution
How to find t*
Can also use invT on the calculator!
Need upper t* value with 5% is above –
0 Use Table B forso
t distributions
95% is below
0 Look up confidence level at bottom &amp; df on the
sides
0 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
0 Have an SRS from population
0  unknown
0 Normal distribution
0 Given
0 Large sample size
0 Check graph of data
Robust
0 An inference procedure is ROBUST if the
confidence level or p-value doesn’t change much if
the assumptions are violated.
0 t-procedures can be used with some skewness, as
long as there are no outliers.
0 Larger n can have more skewness.
0 Outliers are always a concern, but they are even
more of a concern for confidence intervals using
the t-distribution
0 Sample mean is not resistant; hence the sample
mean is larger or smaller (drawn toward the
outlier)
(small numbers of n in t-distribution!)
0 Sample standard deviation is not resistant; hence
the sample standard deviation is larger
0 Confidence intervals are much wider with an outlier
included
0 Options:
0 Make sure data is not a typo (data entry error)
0 Increase sample size beyond 30 observations
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.
We are 95% confident that the true mean
pulse rates of adults is between 70.883 and
74.497 beat per minute.
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.
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.
We are 98% confident that the true mean calorie
content per serving is between 126.16 and 189.56
calories.
A diet guide claims that you will get 120 calories from a
serving of vanilla yogurt. What does this evidence
indicate?
Note: confidence intervals tell us
if something is NOT EQUAL –
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:
0The data MUST be a SRS from the
population
0The formula is not correct for
more complex sampling designs,
i.e., stratified, etc.
0No way to correct for bias in data
Cautions continued:
0Outliers can have a large effect on
confidence interval
0Must know  to do a z-interval –
which is unrealistic in practice
0Homework:
010.27, 28, 29
0Pg.648-649
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