Download Confidence Intervals

Confidence
Intervals with
Means
Chapter 12 Section 2
What is the purpose of a
confidence interval?
To estimate an unknown
population parameter
In this case we will be estimating
mean.
Steps for doing a confidence
interval:
1) Assumptions –
2) Calculate the interval
3) Write a statement about the interval
in the context of the problem.
Assumptions for z-inference
• Have an SRS from population (or
randomly assigned treatments)
• s known
• Normal (or approx. normal) distribution
– Given
– Large sample size
Use only one of
these methods to
• Population is at least 10ncheck normality
Formula:
Standard
deviation of
Critical value
statistic
Confidence Interval :
 s 

x  z * 
 n
statistic
Margin of error
Statement: (memorize!!)
We are ________% confident
that the true mean context is
between ______ and ______.
Example
A sample of pulse rates of 40
women selected at random had a
mean of 76.3 per minute. If the
standard deviation of the
population is 12.5 use a .95
confidence level to find a
confidence interval for .
Determining Sample Size
You want to estimate the mean
number of sentences in a
magazine ad. How many
magazine ads must be included in
the sample if you want to be 95%
confident that the sample mean is
within one sentence of the
population mean? (use s = 5.0)
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. Find a 95% confidence
interval for the true mean systolic blood
pressure of the placebo group.
Can you find a z-interval for this
problem? Why or why not?
Student’s t- distribution
In many real-life situations, the
population standard deviation is
not known and it is often not
practical to collect sample sizes of
30 or more.
However, if the random variable is
approximately normally
distributed, the sampling
distribution for x is a t-distribution.
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
df = n - 1
Graph examples of
t- curves vs standard normal curve
How does the t-distributions
compare to the standard
normal distribution?
• Shorter & more spread out
• More area under the tails
• As n increases, t-distributions
become more like a standard
normal distribution
Steps for doing a confidence
interval:
1) Assumptions –
2) Calculate the interval
3) Write a statement about the interval
in the context of the problem.
Assumptions for t-inference
• Have an SRS from population (or
randomly assigned treatments)
• s unknown
• Normal (or approx. normal) distribution
– Given
– Large sample size
– Check graph of data
Use only one of
these methods to
check normality
Formula:
Standard
deviation of
Standard error
–
Critical value
statistic
when you
substitute s for s.
Confidence Interval :
 s 

x  t * 
 n
statistic
Margin of error
How to find t*
Can also use invT on the calculator!
• Use Table B 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
Statement: (memorize!!)
We are ________% confident
that the true mean context is
between ______ and ______.
CI & p-values deal with area in the tails
Robust
– is the area changed greatly when
there
is
skewness
• An inference procedure is ROBUST if
the confidence level or p-value doesn’t
change much if the normality
assumption is violated.
Since there is more area in the tails in tdistributions,can
then,
a distribution
has
• t-procedures
beif used
with some
some skewness,
tail area
not
skewness,
as long the
as there
areisno
greatly affected.
outliers.
• Larger n can have more skewness.
In a randomized comparative experiment
on the effects of calcium on blood
pressure, researchers divided 54 healthy,
white males at random into two groups,
each person takes calcium or a 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. Find a 95%
confidence interval for the true mean
systolic blood pressure of the placebo
group.
Ex. 1) Find a 95% confidence interval for the
true mean systolic blood pressure of the
placebo group.
Assumptions:
• Have randomly assigned males to treatment
• Systolic blood pressure is normally distributed (given).
• s is unknown
•Population is at least 270 males
 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.
Ex. 2) 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 rate of adults is
between 70.883 & 74.497. (Be
sure to include assumptions and
Calculations!)
Ex 2 continued) 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. 3) 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 of vanilla
yogurt is between 126.16 calories & 189.56
calories.
Ex 3 continued) A diet guide claims that
you will get 120 calories from a serving
Note: confidence intervals tell us
of vanilla yogurt. What does this
if something is NOT EQUAL –
evidence
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.
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:
 s 
m  z *

 n
Always round up to the nearest person!
Ex 4) The heights of MHS male
students is normally distributed with
s = 2.5 inches. How large a sample
is necessary to be accurate within +
.75 inches with a 95% confidence
interval?
n = 43
Some Cautions:
• The data MUST be a SRS from the
population (or randomly assigned
treatment)
• 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 s to do a z-interval –
which is unrealistic in practice